Almost periodic and asymptotically almost periodic functions: part I
- Dhaou Lassoued^{1, 2},
- Rahim Shah^{3} and
- Tongxing Li^{4, 5}Email author
https://doi.org/10.1186/s13662-018-1487-0
© The Author(s) 2018
Received: 18 October 2017
Accepted: 12 January 2018
Published: 2 February 2018
Abstract
In this paper, we review indispensable properties and characterizations of almost periodic functions and asymptotically almost periodic functions in Banach spaces. Special accent is put on the Stepanov generalizations of almost periodic functions and asymptotically almost periodic functions. We also recollect some basic results regarding equi-Weyl-almost periodic functions and Weyl-almost periodic functions. The class of asymptotically Weyl-almost periodic functions, introduced in this work, seems to be not considered elsewhere even in the scalar-valued case. We actually introduce eight new classes of asymptotically almost periodic functions and analyze relations between them. In order to make a picture as complete and clear as possible, several illustrating examples and counter-examples are given. It is worth noting that the topics dealt with in this paper seem to be of an intrinsic connection with the problem of existence and uniqueness of solutions of differential and difference equations, in both determinist and stochastic cases.
Keywords
MSC
1 Introduction
The theory of almost periodic functions has gradually been increased to a comprehensive and extensive theory by the contributions of numerous mathematicians. Indeed, the prehistory of almost periodicity begins with Esclangon and Bohl. The theory of almost periodic functions was developed in its main features by Bohr as a generalization of pure periodicity in three rather long papers [1–3], under the common title ‘Zur Theorie der fastperiodischen Funktionen’ in 1925 and 1926. The first of these papers dealt with the almost periodic functions of a real variable, while the third one took up the case of a complex variable. Afterwards, the theory of almost periodic functions was continuously getting established by several mathematicians like Amerio and Prouse [4], Levitan [5], Besicovitch, Bochner, von Neumann, Fréchet, Pontryagin, Lusternik, Stepanov, Weyl, etc.; with respect to this matter, we cite [6–10] and the references therein. In 1962, Bochner [11] defined and studied the almost periodic functions with values in Banach spaces. He showed that these functions include certain earlier generalizations of the notion of almost periodic functions. Some extensions of Bohr’s concept have been introduced, most notably by Besicovitch, Stepanov, Weyl and Eberlein. One can remark that speaking about Stepanov, Weyl or Besicovitch metrics implicitly means dealing with the related quotient spaces, because otherwise we should rather speak about Stepanov, Weyl or Besicovitch.
In fact, the first motivation for the study of almost periodic functions is the set of various ways to combine periodic functions with different periods. For instance, the function \(x \mapsto \cos x + \cos (5x)\) is periodic, and this remains true when 5 is replaced by any other rational number. However, the sum of the periodic functions \(x \mapsto e^{ix}\) and \(x \mapsto e^{i\sqrt{2} x}\) is not periodic. Hence, when such functions, obtained by using a combination of periodic functions, are not periodic, they are not without properties: they are almost periodic functions. In the courses of mechanics, we usually encounter some two-dimensional differential systems of the form \(x' = Ax+e(t)\), where A is a \(2\times 2\) matrix with purely imaginary eigenvalue and \(e(\cdot )\) is a periodic exterior force. It is well known that when these forced systems possess a periodic oscillation, then the period of this oscillation is exactly the period of the exterior force. It is not mentioned in these courses, but these forced systems possess almost periodic solutions. More generally, we know that when all the solutions of an autonomous linear finite dimensional system are bounded, then all these solutions are almost periodic. In more physical terms, the almost periodic trajectories are trajectories with a discrete spectrum. Besides, among the actual literature about the chaos theory, a famous model of transition towards the chaos is the Landau-Hopf model [12] where the involved potential is an almost periodic potential. Maurice Allais (Nobel Price of Economics) has written a wide work about the foundations of the theory of probabilities [13]. The major conclusion of his work is the following: many natural phenomena are considered as stochastic phenomena, but, in fact, they are almost periodic phenomena which are badly understood. In support of his viewpoint, Allais has established (with rigorous proofs [13]) a mathematical theorem which says that the samplings of an almost periodic function converge to the Laplace-Gauss distribution.
Ever since their introduction by Bohr in the mid-twenties, almost-periodic (a.p.) functions have played an important role in various branches of mathematics. Also, in the course of time, various variants and extensions of Bohr’s concept have been introduced, most notably by Besicovitch, Stepanov and Weyl. Accordingly, there are a number of monographs and papers covering a wide spectrum of notions of almost periodicity and applications (see, for instance, the large list of references [14, Chapters 1 and 2]). An extension of Bohr’s original (scalar) concept of a different kind is the generalization to vector-valued almost-periodic functions, starting with Bochner’s work in the thirties. Here, too, are a number of monographs on the subject, most notably by Amerio and Prouse [4] and Levitan and Zhikov [15]. This vector-valued (Banach space-valued) case is particularly important for applications to (the asymptotic behavior of solutions to) differential equations and dynamical systems.
As aforementioned, the notion and properties of almost periodic functions, either in their initial or in generalized form, turned out to be of great importance in various fields of analysis, function theory, topology and applied mathematics. The necessity of a manuscript giving a concise and systematic exposition of the fundamentals of the theory of almost periodic functions was becoming more and more obvious. The task of writing such a manuscript in a just only one part was an arduous one. Therefore, it is not astonishing that the present article will lead to other future works running in the same aim.
In the present article, we study the basic properties of almost periodic functions and asymptotically almost periodic functions. These topics are intrinsically connected with the problem of existence and uniqueness of solutions of differential equations. To give a complete and clear picture for the different spaces studied in our work, we illustrate them by several examples and counter-examples.
Throughout this paper, we use the usual notation \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{R}\) and \(\mathbb{C}\) for the sets of natural, integer, real and complex numbers, respectively.
For any real number \(s \in \mathbb{R}\), we denote \(\lfloor s \rfloor = \sup \{ l\in \mathbb{Z}: s\geq l \}\) and \(\lceil s \rceil = \inf \{l \in \mathbb{Z}: s \leq l \}\).
Unless stated otherwise, we assume that X is an infinite-dimensional complex Banach space, and the norm of an element \(x\in X\) is denoted by \(\Vert x \Vert \). Assuming that Y is another complex Banach space, we denote by \(L(X, Y)\) the space consisting of all continuous linear mappings from X into Y and \(L(X)\equiv L(X,X)\). The norm on \(L(X, Y)\) shall be denoted by the same notation \(\Vert \cdot \Vert \). The topology on \(L(X, Y)\) and \(X^{*}:=L(X,\mathbb{C})\), the dual space of X, are introduced in the usual way. The symbol I denotes the identity operator on X. In some places, we need to have two different pivot spaces, thus we sometimes use the symbols \(Y, Z ,\ldots, E\) in place of X.
Let \(I = \mathbb{R}\) or \(I = \mathopen[0, \infty\mathclose)\). The space of all Bochner integrable functions from I into X is denoted by \(L^{1}(I : X)\), equipped with the norm \(\Vert f \Vert _{1} = \int_{I} \Vert f(t) \Vert \,dt \). For \(1 \leq p < \infty \) and \((\Omega , \mathcal{R}, \mu )\) a measure space, by \(L^{p}(\Omega : X)\) we denote the space consisting of all strongly μ-measurable functions \(f :\Omega \rightarrow X\) such that \(\Vert f \Vert _{p} := ( \int_{\Omega } \Vert f(\cdot ) \Vert ^{p} \,d\mu )^{1/p}\) is finite. By \(C_{b}(I : X)\) we denote the space consisting of all bounded continuous functions from I into X. The symbol \(C_{0}(\mathopen[0, \infty\mathclose) : X)\) denotes the closed subspace of \(C_{b}(I : X)\) consisting of functions vanishing as the module of the argument tends to infinity. By \(\operatorname {BUC}(I : X)\) we denote the space consisting of all bounded uniformly continuous functions from I to X. The sup-norm turns these spaces into Banach’s. The notation \(c_{0}\) will be deserved to the space of all complex sequences \((a_{n})_{n}\) that converge to zero at infinity, that is, such that \(\lim_{n\rightarrow \infty } \vert a_{n} \vert =0\). Sometimes, we use the notation \(X^{I}\) for the set of all applications from I into X. For an application \(f \in X^{I}\), \(R(f)\) denotes its range (or its image).
Our paper is organized in three big sections. In the first section, we study general almost periodic functions and asymptotically almost periodic ones. In the second section, we deal with the Stepanov generalization for almost periodicity and asymptotic almost periodicity. Then, Weyl almost periodic functions and asymptotically almost periodic functions are considered. These sections are mutually closely connected since we analyze some comparison relations linking the different functional spaces defined in each paragraph.
2 Almost periodic functions and asymptotically almost periodic functions
As underlined above, the concept of almost periodicity was introduced by Danish mathematician Bohr around 1924-1926 and later generalized by many other authors (cf. [10, 15–20] for more details on the subject).
The set consisting of all ε-periods for \(f(\cdot )\) is denoted by \(\mathcal{V}(f, \varepsilon )\).
It is said that the function \(f(\cdot )\) is almost periodic, a.p. for short, if and only if, for each \(\varepsilon >0\), the set \(\mathcal{V}(f, \varepsilon )\) is relatively dense in I, which means that there exists a constant \(l >0\) such that any subinterval of I of length l meets \(\mathcal{V}(f, \varepsilon )\).
Since for each \(\varepsilon >0\) we have \(\mathcal{V}(f, \varepsilon ) \subset \mathcal{V}( \Vert f \Vert , \varepsilon ) \), which is a consequence of the inequality \(\vert \Vert x \Vert - \Vert y \Vert \vert \leq \Vert x - y \Vert \), \(x,y \in X\), it immediately follows from the definition that the almost periodicity of function \(f : I \rightarrow X\) implies the almost periodicity of scalar-valued function \(\Vert f \Vert : I \rightarrow \mathbb{R}\). Furthermore, it can be easily seen that the almost periodicity of function \(f : I \rightarrow X\) implies the almost periodicity of vector-valued functions \(f(\cdot + a)\) and \(f(a\cdot )\), where \({a \in I}\).
We call \(f(\cdot )\) weakly almost periodic, w.a.p. for short, if and only if for each \(x^{*} \in X^{*}\) the function \(x^{*}(f(\cdot ))\) is almost periodic (it is well known that any function \(f \in \operatorname {BUC}(I : X)\) which has a relatively compact range in X and which is w.a.p., needs to be a.p., cf. [21, Proposition 4.5.12]). A family of functions \(\mathcal{F} \subset X^{I}\) is said to be uniformly almost periodic if and only if, for each \(\varepsilon >0\), there exists a constant \(l > 0\) such that any subinterval of I of length l contains a number \(\tau > 0\) such that (2.1) holds for all \(f \in \mathcal{F}\).
The space consisting of all almost periodic functions from the interval I into X will be denoted by \(\operatorname {AP}(I : X)\). Equipped with the sup-norm, \(\operatorname {AP}(I : X)\) becomes a Banach space.
The most intriguing properties of almost periodic vector-valued functions are collected in the following two theorems (in the case that \(I = \mathbb{R}\) these assertions are well known in the existing literature, in the case that \(I = \mathopen[0, \infty\mathclose)\), then these assertions can be deduced by using their validity in the case \(I = \mathbb{R}\) and the properties of extension mapping \(E(\cdot )\)).
Theorem 2.1
- (1)
\(f \in \operatorname {BUC}(I : X)\);
- (2)
if \(g \in \operatorname {AP}(I : X)\), \(h \in \operatorname {AP}(I : \mathbb{C})\), \(\alpha , \beta \in \mathbb{C}\), then \(\alpha f + \beta g\) and hf belong to \(\operatorname {AP}(I : X)\);
- (3)Bohr’s transform of \(f(\cdot )\)exists for all \(r\in \mathbb{R}\) and$$ P_{r}(f):= \lim_{t\rightarrow \infty } \frac{1}{t} \int_{0}^{t} e^{-irs} f(s) \,ds $$for all \(\alpha \in I\) and \(r\in \mathbb{R}\);$$ P_{r}(f) = \lim_{t\rightarrow \infty } \frac{1}{t} \int_{\alpha }^{t + \alpha } e^{-irs} f(s) \,ds $$
- (4)
if \(P_{r}(f) = 0\) for all \(r \in \mathbb{R}\), then \(f(t) = 0\) for all \(t \in I\);
- (5)
\(\sigma (f) := \{r \in \mathbb{R} : P_{r}(f) \neq 0\}\) is at most countable;
- (6)
if \(c_{0} \nsubseteq X\), which means that X does not contain an isomorphic copy of \(c_{0}\), \(I = \mathbb{R}\) and \(g(t) = \int_{0}^{t} f(s) \,ds\) (\(t \in \mathbb{R}\)) is bounded, then \(g \in \operatorname {AP}(\mathbb{R} : X)\);
- (7)
if \((g_{n})_{n\in \mathbb{N}}\) is a sequence in \(\operatorname {AP}(I : X)\) and \((g_{n})_{n\in \mathbb{N}}\) converges uniformly to g, then \(g \in \operatorname {AP}(I : X)\);
- (8)
if \(I = \mathbb{R}\) and \(f' \in \operatorname {BUC}(\mathbb{R} : X)\), then \(f' \in \operatorname {AP}( \mathbb{R} : X)\);
- (9)
(spectral synthesis) \(f \in \overline{ \operatorname {span}\{e^{i\mu } x : \mu \in \sigma (f), x \in R(f) \}}\);
- (10)
\(R(f)\) is relatively compact in X;
- (11)we have$$ \Vert f \Vert _{\infty }= \sup_{t\geq t_{0}} \bigl\Vert f(t) \bigr\Vert , \quad t_{0} \in I; $$(2.2)
- (12)
if \(I = \mathbb{R}\) and \(g \in L^{1}(\mathbb{R})\), then \(g \star f \in \operatorname {AP}(\mathbb{R} : X)\), where \((g \star f )(t)= \int_{-\infty }^{\infty }g(t-s)f(s)\,ds\), \(t\in \mathbb{R}\).
Theorem 2.2
(Bochner’s criterion)
Let \(f \in \operatorname {BUC}(\mathbb{R} : X)\). Then \(f(\cdot )\) is almost periodic if and only if, for any sequence \((b_{n})_{n}\) of numbers from \(\mathbb{R}\), there exists a subsequence \((a_{n})_{n}\) of \((b_{n})_{n}\) such that \((f(\cdot + a_{n}))_{n}\) converges in \(\operatorname {BUC}(R : X)\).
Remark 2.3
Before proceeding any further, we would like to mention that the necessary and sufficient condition for X to contain \(c_{0}\) is given in [21, Theorem 4.6.14] : \(c_{0} \subseteq X\) if and only if there exists a divergent series \(\sum_{n=1}^{\infty }x_{n}\) in X which is unconditionally bounded, i.e., there exists a constant \(M > 0\) such that \(\Vert \sum_{j=1}^{m} x_{n_{j}} \Vert \leq M\), whenever \(n_{j} \in \mathbb{N}\) (\(j = 1, 2,\ldots, m\)) such that \(n_{1} < n_{2} < \cdots < n_{m}\). The importance of condition \(c_{0} \nsubseteq X\) has been recognized already by Bohr and later employed by many others (see, e.g., Kadet’s theorem [21, Theorem 4.6.11]).
By either \(\operatorname {AP}(\Lambda : X)\) or \(\operatorname {AP}_{\Lambda }(I : X)\), where Λ is a nonempty subset of I, we denote the vector subspace of \(\operatorname {AP}(I : X)\) consisting of all functions \(f \in \operatorname {AP}(I : X)\) for which the inclusion \(\sigma (f) \subseteq \Lambda \) holds good. It can be easily seen that \(\operatorname {AP}(\Lambda : X)\) is a closed subspace of \(\operatorname {AP}(I : X)\) and therefore Banach space itself.
The relative compactness of subsets in \(\operatorname {AP}(I : X)\) has been examined by Corduneanu [23] (see also [17, Theorem 3.11]). A function \(f \in \operatorname {BUC}(I : X)\) is said to be weakly almost periodic in the sense of Eberlein if and only if \(\{f(\cdot + s) : s \in I \}\) is relatively weakly compact in X. This important class of functions will not be considered in the sequel (for further details concerning this intriguing topic and connections between almost periodicity and Carleman spectrum of functions, one may refer to the monograph [21] and the references cited therein).
2.1 Asymptotically almost periodic functions
The notion of an asymptotically almost periodic function was introduced by Fréchet in 1941 (for more details concerning the vector-valued asymptotically almost periodic functions and asymptotically almost periodic differential equations, see, e.g., [17, 18, 24–30]).
A function \(f \in C_{b}(\mathopen[0, \infty\mathclose) : X)\) is said to be asymptotically almost periodic if and only if, for every \(\varepsilon > 0\), we can find numbers \(l > 0\) and \(M > 0\) such that every subinterval of \(\mathopen[0, \infty\mathclose)\) of length l contains, at least, one number τ such that \(\Vert f(t + \tau ) - f(t) \Vert \leq \varepsilon \) for all \(t \geq M\). The space consisting of all asymptotically almost periodic functions from \(\mathopen[0, \infty\mathclose)\) into X will be denoted by \(\operatorname {AAP}(\mathopen[0, \infty\mathclose) : X)\).
- (i)
\(f \in \operatorname {AAP}(\mathopen[0, \infty\mathclose) : X)\);
- (ii)
there exist uniquely determined functions \(g \in \operatorname {AP}(\mathopen[0, \infty\mathclose) : X)\) and \(\Phi \in C_{0}(\mathopen[0, \infty\mathclose) : X)\) such that \(f = g + \Phi\);
- (iii)
the set \(H(f) := \{f(\cdot +s) : s > 0\}\) is relatively compact in \(C_{b}(\mathopen[0, \infty\mathclose) : X)\), which means that for any sequence \((b_{n})_{n}\) of nonnegative real numbers there exists a subsequence \((a_{n})_{n}\) of \((b_{n})_{n}\) such that \((f(\cdot+a_{n}))_{n}\) converges in \(C_{b}(\mathopen[0, \infty\mathclose) : X)\).
The functions g and Φ from (ii) are called the principal and corrective terms of the function f, respectively. Then we know that \(\overline{ R(g)} \subseteq \overline{ R(f)}\) (see, e.g., [17, Lemma 3.43]).
By \(C_{0}(\mathopen[0, \infty\mathclose) \times Y : X)\), we denote the space of all continuous functions \(h : \mathopen[0, \infty\mathclose)\times Y \rightarrow X\) such that \(\lim_{t\rightarrow 0} h(t, y) = 0\) uniformly for y in any compact subset of Y. A continuous function \(f : I \times Y \rightarrow X\) is called uniformly continuous on bounded sets, uniformly for \(t \in I\) if and only if, for every \(\varepsilon > 0\) and every bounded subset K of Y, there exists a number \(\delta_{\varepsilon , K} > 0\) such that \(\Vert f(t, x) - f(t, y) \Vert \leq \varepsilon \) for all \(t \in I\) and all \(x, y \in K \) satisfying that \(\Vert x - y \Vert \leq \delta_{\varepsilon , K}\). If \(f : I \times Y \rightarrow X\), then we define \(\hat{f} : I \times Y \rightarrow L^{p}([0, 1] : X)\) by \(\hat{f(t, y)} := f(t + \cdot , y)\), \(t \geq 0\), \(y \in Y\). For the purpose of research of (asymptotically) almost periodic properties of solutions to semilinear Cauchy inclusions, we need to remind ourselves of the following well-known definitions and results (see, e.g., Zhang [34], Long and Ding [35] and Proposition 2.6 below).
Definition 2.4
- (1)
A function \(f : I \times Y \rightarrow X\) is called almost periodic if and only if \(f(\cdot ,\cdot )\) is bounded, continuous as well as, for every \(\varepsilon > 0\) and every compact \(K \subset Y\), there exists an \(l(\varepsilon , K) > 0\) such that every subinterval \(J \subset I\) of length \(l(\varepsilon , K)\) contains a number τ with the property that \(\Vert f(t + \tau , y) - f(t, y )\Vert \leq \varepsilon \) for all \(t \in I\), \(y \in K\). The collection of such functions will be denoted by \(\operatorname {AP}(I \times Y : X)\).
- (2)
A function \(f : \mathopen[0, \infty\mathclose) \times Y \rightarrow X\) is said to be asymptotically almost periodic if and only if it is bounded continuous and admits a decomposition \(f = g + q\), where \(g \in \operatorname {AP}(\mathopen[0, \infty\mathclose) \times Y : X)\) and \(q \in C_{0}(\mathopen[0, \infty\mathclose)\times Y : X)\). Denote by \(\operatorname {AAP}(\mathopen[0, \infty\mathclose) \times Y : X)\) the vector space consisting of all such functions.
The following composition principles are well known in the existing literature (see, e.g., [34]).
Theorem 2.5
- (1)
Let \(f \in \operatorname {AP}(I \times Y : X)\) and \(h \in \operatorname {AP}(I : Y )\). Then the mapping \(t \mapsto f(t, h(t))\), \(t \in I\), belongs to the space \(\operatorname {AP}(I : X)\).
- (2)
Let \(f \in \operatorname {AAP}(\mathopen[0, \infty\mathclose) \times Y : X)\) and \(h \in \operatorname {AAP}(\mathopen[0, \infty\mathclose) : Y )\). Then the mapping \(t \mapsto f(t, h(t))\), \(t \geq 0\), belongs to the space \(\operatorname {AAP}(\mathopen[0, \infty\mathclose) : X)\).
In Definition 2.4(2), a great number of authors assume a priori that \(g \in \operatorname {AP}(\mathbb{R}\times Y : X)\). This is slightly redundant on account of the following proposition.
Proposition 2.6
Let \(f : \mathopen[0, \infty\mathclose) \times Y \rightarrow X\) and let \(S \subseteq Y\). Suppose that, for every \(\varepsilon > 0\), there exists an \(l(\varepsilon , S) > 0\) such that every subinterval \(J \subseteq \mathopen[0, \infty\mathclose)\) of length \(l(\varepsilon , S)\) contains a number τ with the property that \(\Vert f(t+ \tau , y) - f(t, y) \Vert \leq \varepsilon \) for all \(t \geq 0\), \(y \in S\) (this, in particular, holds provided that \(f \in \operatorname {AP}(I \times Y : X)\)).
Denote by \(F(t, y)\) the unique almost periodic extension of function \(f(t, y)\) from the interval \(\mathopen[0, \infty\mathclose)\) to the whole real line for fixed \(y \in S\).
Then, for every \(\varepsilon > 0\), with the same \(l(\varepsilon , S) > 0\) chosen as above, we have that every subinterval \(J \subseteq \mathbb{R}\) of length \(l(\varepsilon , S)\) contains a number τ with the property that \(\Vert F(t+ \tau , y) - F(t, y) \Vert \leq \varepsilon \) for all \(t \in \mathbb{R}\), \(y \in S\).
Proof
Let \(\varepsilon > 0\) be given in advance, \(l(\varepsilon , S) > 0\) be as above, and let \(J = [a, b] \subseteq \mathbb{R}\). The assertion is clear provided that \(a > 0\). Suppose now that \(a < 0\). We choose a number \(\tau_{0} > 0\) arbitrarily. Then there exists a \(\tau '\in J = [\tau_{0}, \tau_{0} + b -a] \subseteq \mathopen[0, \infty\mathclose)\) such that \(\Vert f(t + \tau_{0}, y) - f(t, y) \Vert \leq \varepsilon \) for all \(t \geq 0\), \(y \in S\).
Since \(\tau := \tau ' - \tau_{0} - \vert a \vert \in J\), it suffices to show that \(\Vert F(t + \tau , y) - F(t, y) \Vert \leq \varepsilon \) for all \(t \in \mathbb{R}\), \(y \in S\).
3 Stepanov almost periodic functions and asymptotically Stepanov almost periodic functions
Let \(1 \leq p < \infty \), \(l > 0\), and \(f, g \in L^{p}_{\mathrm {loc}}(I : X)\), where \(I = \mathbb{R}\) or \(I = \mathopen[0, \infty\mathclose)\).
The distance appearing in (3.3) is called the Weyl distance of \(f(\cdot )\) and \(g(\cdot )\).
3.1 Asymptotically Stepanov almost periodic functions
It is said that \(f \in L^{p}_{S}(\mathopen[0, \infty\mathclose) : X)\) is asymptotically Stepanov p-almost periodic, asymptotically \(S^{p}\)-almost periodic shortly, if and only if \(\hat{f} : \mathopen[0, \infty\mathclose) \rightarrow L^{p}([0, 1] : X)\) is asymptotically almost periodic.
It is a well-known fact that if \(f(\cdot )\) is an almost periodic (respectively, a.p.) function, then \(f(\cdot )\) is also \(S^{p}\)-almost periodic (respectively, \(S^{p}\)-a.p.) for \(1\leq p < \infty \). The converse statement is false, however, as the following example from the book of Levitan [5] shows.
Example 3.1
Hereafter, we will also use the Bochner theorem, which asserts that any BUC function that is Stepanov p-almost periodic needs to be almost periodic (\(1 \leq p < \infty \)).
The notion of a scalar \(S^{p}\)-almost periodic function, slightly different from the notion of usually considered weakly \(S^{p}\)-almost periodic function, is given as follows: a function \(f \in L^{p}_{S}(I : X)\) is said to be scalarly Stepanov p-almost periodic if and only if, for each \(x^{*} \in X^{*}\), we have that the function \(x^{*}(f): \mathopen[0, \infty\mathclose) \rightarrow \mathbb{C}\) defined by \(x^{*}(f)(t):= x^{*}(f(t))\), \(t\geq 0\), is Stepanov p-almost periodic.
Definition 3.2
A function \(f : I \times Y \rightarrow X\) is called Stepanov p-almost periodic, \(S^{p}\)-almost periodic shortly, if and only if \(\hat{f}: I \times Y \rightarrow L^{p}([0, 1] : X)\) is almost periodic.
By [34, Theorem 2.6], we have that a bounded continuous function \(f : \mathopen[0, \infty\mathclose)\times Y \rightarrow X\) is asymptotically almost periodic if and only if, for every \(\varepsilon > 0\) and every compact \(K \subseteq Y\), there exist \(l(\varepsilon , K) > 0\) and \(M(\varepsilon , K) > 0\) such that every subinterval \(J \subseteq \mathopen[0, \infty\mathclose)\) of length \(l(\varepsilon , K)\) contains a number τ with the property that \(\Vert f(t+\tau , y) - f(t, y) \Vert \leq \varepsilon \) for all \(t > M(\varepsilon , K)\), \(y \in K\).
We introduce the notion of an asymptotically Stepanov p-almost periodic function \(f(\cdot , \cdot )\) as follows.
Definition 3.3
Let \(1 \leq p <\infty \). A function \(f : \mathopen[0, \infty\mathclose) \times Y \rightarrow X\) is said to be asymptotically \(S^{p}\)-almost periodic if and only if \(\hat{f}: \mathopen[0, \infty\mathclose) \times Y \rightarrow L^{p}([0, 1] : X)\) is asymptotically almost periodic. The collection of such functions will be denoted by \(\operatorname {AAPS}^{p}(\mathopen[0, \infty\mathclose)\times Y : X)\).
It is very elementary to prove that any asymptotically almost periodic function is also asymptotically Stepanov p-almost periodic (\(1 \leq p < \infty \)).
We need the assertion of [36, Lemma 1].
Lemma 3.4
- (1)
g is \(S^{p}\)-almost periodic;
- (2)
q̂ belongs to the class \(C_{0}([0, \infty): L^{P}([0, 1]: X))\);
- (3)
\(f(t)= g(t) + q(t)\) for all \(t \geq 0\).
Now we state the following two-variable analogue of Lemma 3.4.
Lemma 3.5
- (1)
\(\hat{g}: \mathbb{R} \times Y \rightarrow L^{p}([0, 1]: X) \) is almost periodic;
- (2)
q̂ belongs to the class \(C_{0}(\mathopen[0, \infty\mathclose)\times Y : L^{P}([0, 1]: X))\);
- (3)
\(f(t,y)= g(t,y) + q(t,y)\) for all \(t \geq 0\) and \({y\in Y}\).
Proof
By the foregoing, we have that \(\hat{f} : \mathopen[0, \infty\mathclose)\times Y \rightarrow X\) is bounded continuous and admits a decomposition \(\hat{f} = G+Q\), where \(G \in \operatorname {AP}(\mathopen[0, \infty\mathclose)\times Y : L^{p}([0, 1] : X))\) and \(Q \in C_{0}(\mathopen[0, \infty\mathclose)\times Y : L^{p}([0, 1] : X))\). Moreover, the proof of [34, Theorem 2.6] shows that, for every compact set \(K \subseteq Y\), there exists an increasing sequence \((t_{n})_{n \in \mathbb{N}}\) of positive reals such that \(\lim_{n \rightarrow \infty } t_{n} = \infty \) and \(G(t, y) = \lim_{n \rightarrow \infty } \hat{f}(t+ t_{n}, y)\) for all \(y\in Y\) and \(t\geq 0\). The remaining part of proof follows by applying Lemma 3.4 to the function \(\hat{f}(\cdot , y)\) for a fixed element \(y \in Y\) and the uniqueness of decomposition \(g(\cdot ) + q(\cdot )\) in this lemma. □
In the case that the value of p is irrelevant, we simply say that the function under our consideration is (asymptotically, scalarly) Stepanov almost periodic. Hereafter, we will use the following lemma (see, e.g., [6, p. 70] for the scalar-valued case).
Lemma 3.6
4 Weyl almost periodic functions and asymptotically Weyl almost periodic functions
Unless specified otherwise, in this section it will be always assumed that \(I =\mathbb{R}\) or \(I = \mathopen[0, \infty\mathclose)\). The pivot Banach space will be denoted by X. The notion of an (equi-)Weyl almost periodic function is given as follows (cf. also (3.1)).
Definition 4.1
- (1)We say that the function \(f(\cdot )\) is equi-Weyl-p-almost periodic, \(f \in \mathit {e}\text{-}W^{p}_{\mathrm {ap}}(I : X)\) for short, if and only if, for each \(\varepsilon > 0\), we can find two real numbers \(l > 0\) and \(L > 0\) such that any interval \(I' \subseteq I\) of length L contains a point \(\tau \in I'\) such thatthat is,$$ \sup_{x\in I} \frac{1}{l} \biggl[ \int_{x}^{x+l} \bigl\Vert f(t+ \tau ) - f(t) \bigr\Vert ^{p} \,dt \biggr]^{1/p} \leq \varepsilon , $$$$ D^{p}_{S_{l}} \bigl[ f(\cdot + \tau ), f(\cdot ) \bigr] \leq \varepsilon . $$
- (2)We say that the function \(f(\cdot )\) is Weyl-p-almost periodic, \(f \in W^{p}_{\mathrm {ap}}(I : X)\) for short, if and only if, for each \(\varepsilon > 0\), we can find a real number \(L > 0\) such that any interval \(I' \subseteq I\) of length L contains a point \(\tau \in I'\) such thatthat is,$$ \lim_{l\rightarrow \infty }\sup_{x\in I} \frac{1}{l} \biggl[ \int_{x} ^{x+l} \bigl\Vert f(t+ \tau ) - f(t) \bigr\Vert ^{p} \,dt \biggr]^{1/p} \leq \varepsilon , $$$$ \lim_{l\rightarrow \infty }D^{p}_{S_{l}} \bigl[ f(\cdot + \tau ), f( \cdot ) \bigr] \leq \varepsilon . $$
For instance, the scalar-valued function \(f : \mathbb{R} \rightarrow \mathbb{C}\) defined by \(f(x) := \chi_{(0, 1/2)}(x)\), \(x \in \mathbb{R}\) is not Stepanov 1-almost periodic, but it is equi-Weyl-almost-1-periodic (see, e.g., [37, Example 4.27]); and the scalar-valued function \(f : \mathbb{R} \rightarrow \mathbb{C}\) defined by \(f(x) := \chi_{(0, \infty )}(x)\), \(x \in \mathbb{R}\) is not equi-Weyl-almost-1-periodic, but it is Weyl-almost-1-periodic (see, e.g., [38, Example 1]). Here, \(\chi (\cdot )\) denotes the characteristic function. We also want to point out that the space of scalar-valued functions \(W^{p}_{\mathrm {ap}}(\mathbb{R} : \mathbb{R})\) seems to be defined and analyzed for the first time by Kovanko [39] in 1944 (according to the information given in the survey paper [37]).
In the sequel, we use abbreviations \(\mathit {e}\text{-}W_{\mathrm {ap}}(I : X)\) and \(W_{\mathrm {ap}}(I : X)\) to denote the spaces \(\mathit {e}\text{-}W^{1}_{\mathrm {ap}}(I : X)\) and \(W^{1}_{\mathrm {ap}}(I : X)\), respectively (the case \(p = 1\) will be most important in our further analysis). Similarly, we say that a function is (equi)-Weyl-almost periodic if and only if it is (equi)-Weyl-1-almost periodic.
It is very important to state the following characteristic of the space \(\mathit {e}\text{-}W^{p}_{\mathrm {ap}}(I : X)\), see, e.g., [37] for the scalar-valued case.
Theorem 4.2
A Bochner type theorem holds for Weyl almost periodic functions; see [5, 40].
Theorem 4.3
Let \(1 \leq p < \infty \) and let \(f \in W^{p}_{\mathrm {ap}}(I : X)\) be uniformly continuous. Then \(f \in \operatorname {AP}(I : X)\).
It is well known that the functions belonging to the space \(\mathit {e}\text{-}W^{p} _{\mathrm {ap}}(I : X)\) need to be Weyl uniformly continuous in the following sense (see [6, p. 84]).
Theorem 4.4
For some other notions of Weyl-almost periodicity, like equi-\(W^{p}\)-normality and \(W^{p}\)-normality, we refer the reader to [37, Section 4].
4.1 Asymptotically Weyl almost periodic functions
Definition 4.5
Now, assume that \(q \in L^{p}(\mathopen[0, \infty\mathclose) : X)\). Then, for each \(\varepsilon > 0\), there exists a \(t_{0}(\varepsilon ) > 0\) such that \(\int_{t}^{\infty } \Vert q(s) \Vert ^{p} \,ds \leq \varepsilon^{p}\), \(t\geq t_{0}(\varepsilon )\). In particular, \(\int_{t}^{t+ 1} \Vert q(s) \Vert ^{p} \,ds \leq \varepsilon^{p}\), \(t\geq t_{0}(\varepsilon )\), and the function \(\hat{q} : \mathopen[0, \infty\mathclose) \rightarrow L^{p}([0, 1] : X)\) belongs to the class \(C_{0}(\mathopen[0, \infty\mathclose) : L^{p}([0, 1] : X))\). The converse statement is not true, however, since the scalar-valued function \(q(t) = t^{-1/(2p)}\), \(t > 0\) satisfies that \(\hat{q} \in C_{0}(\mathopen[0, \infty\mathclose) : L^{p}([0, 1] : X))\) and \(q \notin L^{p}(\mathopen[0, \infty\mathclose) : X)\).
As the following simple counter-example shows, the converse statement does not hold in general.
Example 4.6
Again, the converse statement does not hold in general and a Weyl-p-vanishing function need not be equi-Weyl-p-vanishing.
Example 4.7
Before proceeding further, we would like to note that an equi-Weyl-p-vanishing function \(q(\cdot )\) need not be bounded as \(t \rightarrow \infty \).
Example 4.8
Denote by \(W^{p}_{0} (\mathopen[0, \infty\mathclose) : X)\) and \(\mathit {e}\text{-}W^{p}_{0} (\mathopen[0, \infty\mathclose) : X)\) the sets consisting of all Weyl-p-vanishing functions and equi-Weyl-p-vanishing functions, respectively. The symbol \(S^{p}_{0} (\mathopen[0, \infty\mathclose) : X)\) will be used to denote the set of all functions \(q \in L^{p}_{\mathrm {loc}}(\mathopen[0, \infty\mathclose) : X)\) such that \(\hat{q} \in C_{0}(\mathopen[0, \infty\mathclose) : L^{p}([0, 1] : X))\).
By our previous considerations, Examples 4.6 and 4.7, we have the following result.
Theorem 4.9
By the analysis contained in [37, Example 4.27], the function \(f : \mathopen[0, \infty\mathclose)\rightarrow\mathbb{C} \) defined by \(f(t) := \chi_{(0, 1/2)}(t)\), \(t > 0\) is equi-Weyl-almost periodic. Since this function is also in class \(\mathit {e}\text{-}W^{p}_{0} (\mathopen[0, \infty\mathclose) : X)\), we have that the sums defining \(\mathit {e}\text{-}W^{p}_{\mathrm {aap}} (\mathopen[0, \infty\mathclose): X)\), \(\mathit {ee}\text{-}W^{p}_{\mathrm {aap}}(\mathopen[0, \infty\mathclose) : X)\), \(W^{p}_{\mathrm {aap}}(\mathopen[0, \infty\mathclose) : X)\) and \(\mathit {e}\text{-}W^{p}_{\mathrm {aap}}(\mathopen[0, \infty\mathclose) : X)\) are not direct. For the first four spaces \(\operatorname {AAPW}^{p}(\mathopen[0, \infty\mathclose) : X)\), \(\mathit {e}\text{-}\operatorname {AAPW}^{p}(\mathopen[0, \infty\mathclose) : X)\), \(\operatorname {AAPSW}^{p}(\mathopen[0, \infty\mathclose) : X)\) and \(\mathit {e}\text{-}\operatorname {AAPSW}^{p}(\mathopen[0, \infty\mathclose) : X)\), the sums in their definitions are direct, which follow from the following proposition.
Proposition 4.10
Proof
By the foregoing, any inclusion of this diagram can be strict. Furthermore, for any two function spaces A and B belonging this diagram and satisfying additionally that there is no transitive sequence of inclusions connecting either A and B or B and A, we have that \(A\setminus B \neq \emptyset \) and \(B \setminus A \neq \emptyset \) (the diagram can be expanded by constructing the sums of spaces of (equi-)Weyl almost periodic functions with \(S^{p}_{0} (\mathopen[0, \infty\mathclose) : X)\), which will be not examined here).
We refer the reader to the paper [41] by Diagana et al. for more details about the notion of \(S_{p}^{(n)}\)-almost periodicity (the notion of N-almost periodicity is very well explored in the monograph [15] by Levitan and Zhikov. Stepanov cases have not been introduced so far, to the best knowledge of the authors). For an excellent survey of results about various classes of (Stepanov) almost periodic functions and (Stepanov) asymptotically almost periodic functions, we refer the reader to the review paper [37] by Andres et al. (cf. also Andres et al. [38]), already cited multiple times before.
We round off our paper by introducing the following important definition.
Definition 4.11
Let \(I = \mathbb{R}\) or \(I = \mathopen[0, \infty\mathclose)\), \((R(t))_{t \in I} \subseteq L(X)\) be a strongly continuous operator family, and let ⨁ denote any of (asymptotically) almost periodic properties considered above. Then we say that \((R(t))_{t \in I}\) is ⨁ (asymptotically) almost periodic if and only if the mapping \(t \mapsto R(t)x\), \(t \in I\) is ⨁ (asymptotically) almost periodic for all \(x \in X\). It is said that \((R(t))_{t \in I}\) is uniformly almost periodic if and only if the family \(\{R(\cdot )x : \Vert x \Vert \leq 1 \}\) is uniformly almost periodic.
Declarations
Acknowledgements
The authors express their sincere gratitude to the editors and two anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details. This research is supported by NNSF of P.R. China (Grant No. 61503171), CPSF (Grant No. 2015M582091), NSF of Shandong Province (Grant No. ZR2016JL021), DSRF of Linyi University (Grant No. LYDX2015BS001), and the AMEP of Linyi University, P.R. China.
Authors’ contributions
All three authors contributed equally to this work. They all read and approved the final version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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