A Note on Almost Anti-Periodic Functions in Banach Spaces

The main aim of this note is to introduce the notion of an almost anti-periodic function in Banach space. We prove some characterizations for this class of functions, investigating also its relationship with the classes of anti-periodic functions and almost periodic functions in Banach spaces.


Introduction and Preliminaries
As mentioned in the abstract, the main aim of this note is to introduce the notion of an almost anti-periodic function in Banach space as well as to prove some characterizations for this class of functions. Any anti-periodic function is almost anti-periodic, and any almost anti-periodic function is almost periodic. Unfortunately, almost anti-periodic functions do not have a linear vector structure with the usually considered operations of pointwise addition of functions and multiplication with scalars. The main result of paper is Theorem 2.6, in which we completely profile the closure of linear span of almost anti-periodic functions in the space of almost periodic functions. We also prove some other statements regarding almost anti-periodic functions, and introduce the concepts of Stepanov almost anti-periodic functions, asymptotically almost anti-periodic functions and Stepanov asymptotically almost anti-periodic functions. We investigate the almost anti-periodic properties of convolution products, providing also a few elementary examples and applications.
Let (X, · ) be a complex Banach space. By C b ([0, ∞) : X) we denote the space consisting of all bounded continuous functions from [0, ∞) into X; the symbol C 0 ([0, ∞) : X) denotes the closed subspace of C b ([0, ∞) : X) consisting of functions vanishing at infinity. By BU C([0, ∞) : X) we denote the space consisted of all bounded uniformly continuous functions from [0, ∞) to X. This space becomes one of Banach's endowed with the sup-norm.
The concept of almost periodicity was introduced by Danish mathematician H. Bohr around 1924-1926 and later generalized by many other authors (cf. [6]- [9] and [16] for more details on the subject). Let I = R or I = [0, ∞), and let f : I → X be continuous. Given ǫ > 0, we call τ > 0 an ǫ-period for f (·) iff The set constituted of all ǫ-periods for f (·) is denoted by ϑ(f, ǫ). It is said that f (·) is almost periodic, a.p. for short, iff for each ǫ > 0 the set ϑ(f, ǫ) is relatively dense in I, which means that there exists l > 0 such that any subinterval of I of length l meets ϑ(f, ǫ).
The space consisted of all almost periodic functions from the interval I into X will be denoted by AP (I : X). Equipped with the sup-norm, AP (I : X) becomes a Banach space.
For the sequel, we need some preliminary results appearing already in the pioneering paper [2] by H. Bart and S. Goldberg, who introduced the notion of an almost periodic strongly continuous semigroup there (see [1] for more details on the subject). The translation semigroup (W (t)) t≥0 on AP ([0, ∞) : X), given by is a linear surjective isometry and Ef is the unique continuous almost periodic extension of a function f (·) from AP ([0, ∞) : X) to the whole real line. We have The most intriguing properties of almost periodic vector-valued functions are collected in the following two theorems (in the case that I = R, these assertions are well-known in the existing literature; in the case that I = [0, ∞), then these assertions can be deduced by using their validity in the case I = R and the properties of extension mapping E(·); see [14] for more details). (iv) if P r (f ) = 0 for all r ∈ R, then f (t) = 0 for all t ∈ I; (v) σ(f ) := {r ∈ R : P r (f ) = 0} is at most countable; (vi) if c 0 X, which means that X does not contain an isomorphic copy of c 0 , I = R and g(t) = t 0 f (s) ds (t ∈ R) is bounded, then g ∈ AP (R : X); (vii) if (g n ) n∈N is a sequence in AP (I : X) and (g n ) n∈N converges uniformly to g, then g ∈ AP (I : X); (viii) if I = R and f ′ ∈ BU C(R : X), then f ′ ∈ AP (R : X); (ix) (Spectral synthesis) f ∈ span{e iµ· x : µ ∈ σ(f ), x ∈ R(f )}; (x) R(f ) is relatively compact in X; (xi) we have Theorem 1.2. (Bochner's criterion) Let f ∈ BU C(R : X). Then f (·) is almost periodic iff for any sequence (b n ) of numbers from R there exists a subsequence (a n ) of (b n ) such that (f (· + a n )) converges in BU C(R : X).
Theorem 1.2 has served S. Bochner to introduce the notion of an almost automorphic function, which slightly generalize the notion of an almost periodic function [4]. For more details about almost periodic and almost automorphic solutions of abstract Volterra integro-differential equations, we refer the reader to the monographs by T. Diagana [6], G. M. N'Guérékata [9], M. Kostić [14] and M. Levitan, V. V. Zhikov [16].
By either AP (Λ : X) or AP Λ (I : X), where Λ is a non-empty subset of I, we denote the vector subspace of AP (I : X) consisting of all functions f ∈ AP (I : X) for which the inclusion σ(f ) ⊆ Λ holds good. It can be easily seen that AP (Λ : X) is a closed subspace of AP (I : X) and therefore Banach space itself.

Almost Anti-Periodic Functions
In what follows, by ϑ ap (f, ǫ) we denote the set of all ǫ-antiperiods for f (·).
We introduce the notion of an almost anti-periodic function as follows.
Definition 2.1. It is said that f (·) is almost anti-periodic iff for each ǫ > 0 the set ϑ ap (f, ǫ) is relatively dense in I.
Suppose that τ > 0 is an ǫ-antiperiod for f (·). Applying (2.1) twice, we get that Taking this inequality in account, we obtain almost immediately from elementary definitions that f (·) needs to be almost periodic. Further on, assume that f : Since the set {(2k + 1)ω : k ∈ Z} is relatively dense in I, the above implies that f (·) is almost anti-periodic. Therefore, we have proved the following theorem: It is well known that any anti-periodic function f : I → X is periodic since, with the notation used above, we have that f (t + 2kω) = f (t), k ∈ Z \ {0}, t ∈ I. But, the constant non-zero function is a simple example of a periodic function (therefore, almost periodic function) that is neither anti-periodic nor almost anti-periodic.

Example 2.3.
(i) Consider the function f (t) := sin(πt) + sin(πt √ 2), t ∈ R. This is an example of an almost anti-periodic function that is not a periodic function. This can be verified as it has been done by A. S. Besicovitch [3, almost periodic, not almost anti-periodic and not periodic.
We continue by noting the following simple facts. Let f : I → X be continuous, and let ǫ ′ > ǫ > 0. Then the following holds true: Furthermore, the argumentation contained in the proofs of structural results of [3, pp. 3-4] shows that the following holds: Theorem 2.4. Let f : I → X be almost anti-periodic. Then we have: n∈N is a sequence of almost anti-periodic functions and (g n ) n∈N converges uniformly to a function g : I → X, then g(·) is almost anti-periodic.
Concerning products and sums of almost anti-periodic functions, the situation is much more complicated than for the usually examined class of almost periodic functions: Example 2.5.
(i) The product of two scalar almost anti-periodic functions need not be almost anti-periodic. To see this, consider the functions f 1 (t) = f 2 (t) = cos t, t ∈ R, which are clearly (almost) anti-periodic. Then f 1 (t) · f 2 (t) = cos 2 t, t ∈ R, cos 2 (t + τ ) + cos 2 t ≥ cos 2 t, τ, t ∈ R and therefore ϑ ap (f 1 · f 2 , ǫ) = ∅ for any ǫ ∈ (0, 1). (ii) The sum of two scalar almost anti-periodic functions need not be almost anti-periodic, so that the almost anti-periodic functions do not form a vector space. To see this, consider the functions f 1 (t) = 2 −1 cos 4t and f 2 (t) = 2 cos 2t, t ∈ R, which are clearly (almost) anti-periodic. Then Asssume that f 1 + f 2 is almost anti-periodic. Then the above identity implies that the function t → 8 cos 4 t − 3, t ∈ R is almost anti-periodic, as well. This, in particular, yields that for any ǫ ∈ (0, 1) we can find τ ∈ R such that 8 cos 4 (t + τ ) + 8 cos 4 t − 6 ≤ ǫ, t ∈ R.
Plugging t = π, we get that 8 cos 4 τ + 2 ≤ ǫ, which is a contradiction. Finally, we would like to point out that there exists a large number of much simpler examples which can be used for verification of the statement clarified in this part; for example, the interested reader can easily check that the function t → cos t + cos 2t, t ∈ R is not almost anti-periodic.
Assume that f : I → X is almost anti-periodic. Then it can be easily seen that f (· + a) and f (b ·) are likewise almost anti-periodic, where a ∈ I and b ∈ I \ {0}.
Denote now by AN P 0 (I : X) the linear span of almost anti-periodic functions I → X. By Theorem 2.2(i), AN P 0 (I : X) is a linear subspace of AP (I : X). Let AN P (I : X) be the linear closure of AN P 0 (I : X) in AP (I : X). Then, clearly, AN P (I : X) is a Banach space. Furthermore, we have the following result: Theorem 2.6. AN P (I : X) = AP R\{0} (I : X).
Proof. Since the mapping E : AP ([0, ∞) : X) → AP (R : X) is a linear surjective isometry, it suffices to consider the case in which I = R. Assume first that f ∈ AP R\{0} (I : X). By spectral synthesis (see Theorem 1.1(ix)), we have that where the closure is taken in the space AP (R : X). Since σ(f ) ⊆ R \ {0} and the function t → e iµt , t ∈ R (µ ∈ R \ {0}) is anti-periodic, we have that span{e iµ· x : µ ∈ σ(f ), x ∈ R(f )} ⊆ AN P 0 (R : X). Hence, f ∈ AN P (R : X). The converse statement immediately follows if we prove that, for any fixed function f ∈ AN P (R : X), we have that P 0 (f ) = 0, i.e., By almost periodicity of f (·), the limit in (2.2) exists. Hence, it is enough to show that for any given number ǫ > 0 we can find a sequence (ω n ) n∈N of positive reals such that lim n→∞ ω n = ∞ and By definition of almost anti-periodicity, we have the existence of a number l > 0 such that any interval I n = [nl, (n + 1)l] (n ∈ N) contains a number ω n that is antiperiod for f (·). The validity of (2.3) is a consequence of the following computation: finishing the proof of theorem.
Further on, Theorem 2.6 combined with the obvious equality σ(Ef ) = σ(f ) immediately implies that the unique ANP extension of a function f ∈ AN P ([0, ∞) : X) to the whole real axis is Ef (·). As the next proposition shows, this also holds for almost anti-periodic functions: Proposition 2.8. Suppose that f : [0, ∞) → X is almost anti-periodic. Then Ef : R → X is a unique almost anti-periodic extension of f (·) to the whole real axis.
Proof. The uniqueness of an almost anti-periodic extension of f (·) follows from the uniqueness of an almost periodic extension of f (·). It remains to be proved that Ef : R → X is almost anti-periodic. To see this, let ǫ > 0 be given. Then there exists l > 0 such that any interval I ⊆ [0, ∞) of length l contains a number τ ∈ I such that f (s + τ ) + f (s) ≤ ǫ, s ≥ 0. We only need to prove that any interval I ⊆ R of length 2l contains a number τ ∈ I such that Finally, if I = I 1 ∪ I 2 , where I 1 = [a, 0] (a < 0) and I 2 = [0, b] (b > 0), then |a| ≥ l or b ≥ l. In the case that |a| ≥ l, then the conclusion follows similarly as in the previously considered case. If b ≥ l, then the conclusion follows from the computation where τ ∈ I 2 is an ǫ-antiperiod of f (·).
For various generalizations of almost periodic functions, we refer the reader to [14]. In the following definition, we introduce the notion of a Stepanov almost anti-periodic function. It can be easily seen that any almost anti-periodic function needs to be S palmost anti-periodic, as well as that any S p -almost anti-periodic function has to be S p -almost periodic (1 ≤ p < ∞).

Almost Anti-Periodic Properties of Convolution Products
Since almost anti-periodic functions do not form a vector space, we will focus our attention here to the almost anti-periodic properties of finite and infinite convolution product, which is undoubtedly a safe and sound way for providing certain applications to abstract PDEs. Proposition 3.1. Suppose that 1 ≤ p < ∞, 1/p+1/q = 1 and (R(t)) t>0 ⊆ L(X) is a strongly continuous operator family satisfying that M := ∞ k=0 R(·) L q [k,k+1] < ∞. If g : R → X is S p -almost anti-periodic, then the function G(·), given by is well-defined and almost anti-periodic.
Proof. It can be easily seen that, for every t ∈ R, we have G(t) = ∞ 0 R(s)g(t−s) ds.
Since g(·) is S p -almost periodic, we can apply [15,Proposition 2.11] in order to see that G(·) is well-defined and almost periodic. It remains to be proved that G(·) is almost anti-periodic. Let a number ǫ > 0 be given in advance. Then we can find a finite number l > 0 such that any subinterval I of R of length l contains a number τ ∈ I such that t+1 t g(s + τ ) + g(s) p ds ≤ ǫ p , t ∈ R. Applying Hölder inequality and this estimate, similarly as in the proof of above-mentioned proposition, we get that which clearly implies that the set of all ǫ-antiperiods of G(·) is relatively dense in R.
Keeping in mind Proposition 3.1 and the proof of [15, Propostion 2.13], we can simply clarify the following result: Proposition 3.3. Suppose that 1 ≤ p < ∞, 1/p + 1/q = 1 and (R(t)) t>0 ⊆ L(X) is a strongly continuous operator family satisfying that, for every s ≥ 0, we have that Suppose, further, that f : [0, ∞) → X is asymptotically S p -almost anti-periodic as well as that the locally p-integrable functions g : R → X, q : [0, ∞) → X satisfy the conditions from Definition 3.2(ii). Let there exist a finite number M > 0 such that the following holds: Before providing some applications, we want to note that our conclusions from [15, Remark 2.14] and [14, Proposition 2.7.5] can be reformulated for asymptotical almost anti-periodicity.
It is clear that we can apply results from this section in the study of existence and uniqueness of almost anti-periodic solutions of fractional Cauchy inclusion where D γ t,+ denotes the Riemann-Liouville fractional derivative of order γ ∈ (0, 1), and f : R → X satisfies certain properties, and A is a closed multivalued linear operator (see [8] for the notion). Furthermore, we can analyze the existence and uniqueness of asymptotically (S p -) almost anti-periodic solutions of fractional Cauchy inclusion (DFP) f,γ : D γ t u(t) ∈ Au(t) + f (t), t ≥ 0, where D γ t denotes the Caputo fractional derivative of order γ ∈ (0, 1], x 0 ∈ X and f : [0, ∞) → X, satisfies certain properties, and A is a closed multivalued linear operator (cf. [14] for more details). Arguing so, we can analyze the existence and uniqueness of (asymptotically S p -) almost anti-periodic solutions of the fractional Poisson heat equations in the space X := L p (Ω), where Ω is a bounded domain in R n , b > 0, m(x) ≥ 0 a.e. x ∈ Ω, m ∈ L ∞ (Ω), γ ∈ (0, 1) and 1 < p < ∞; see [8] and [14] for further information in this direction.