On Semi-c-Periodic Functions

Department of Mathematics and Computer Sciences, University of Adrar, Adrar, Algeria Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125 Novi Sad, Serbia Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Santiago de Chile, Chile Laboratory of Mathematics, Modeling and Applications (LaMMA), University of Adrar, Adrar, Algeria Faculty of Civil Engineering, Ss. Cyril and Methodius University, Skopje, Partizanski Odredi 24 P.O. Box 560, 1000 Skopje, North Macedonia


Introduction
e notion of periodicity plays a fundamental role in mathematics. A continuous function f: I ⟶ E, where E is a topological space and I � R or I � [0, ∞), is said to be periodic if and only if there exists a real number ω > 0 such that f(x + ω) � f(x) for all x ∈ I. e notion of periodicity has recently been reconsidered by Alvarez et al. [1], who proposed the following notion: a continuous function f: I ⟶ E, where E is a complex Banach space, is said to be (ω, c)-periodic (ω > 0, c ∈ C∖ 0 { }) if and only if f(x + ω) � cf(x) for all x ∈ I. Due to ( [1], Proposition 2.2), we know that a continuous function f: I ⟶ E is (ω, c)-periodic if and only if the function g(·) ≡ c (− ·/ω) f(·) is periodic and g(x + ω) � g(x) for all x ∈ I; here, c (− ·/ω) denotes the principal branch of the exponential function (see also the research articles [2,3] by Alvarez et al., the conference paper [4] by Pinto, where the idea for introduction of (ω, c)-periodic functions was presented for the first time, and [5,6] for some generalizations of the concept of (ω, c)-periodicity).
In the sequel, by E we denote a complex Banach space equipped with the norm ‖ · ‖; C(I: (1) e class of antiperiodic functions, i.e., the class of (− 1)-periodic functions: in this case, any positive real number ω > 0 satisfying f(x + ω) � − f(x), x ∈ I, is said to be an antiperiod of f(·). Any antiperiodic function is periodic, since we can apply the above functional equality twice in order to see i.e., the class of continuous functions for all x ∈ I. e number ω is usually called Bloch period of f(·), the number k is usually called the Bloch wave vector or Floquet exponent of f(·), and in the case that kω � π, the class of Bloch (ω, k)-periodic functions is equal to the class of antiperiodic functions having the number ω as an antiperiod. If the function f(·) is Bloch (ω, k)-periodic, then we inductively obtain f(x + mω) � e imkω f(x) for all x ∈ I and m ∈ N, so that the function f(·) must be periodic provided that kω ∈ Q, but, if kω ∉ Q, then the function f(·) need not be periodic as the following simple counterexample shows: the function is Bloch (ω, k)-periodic with ω � 2π + � 2 √ π and k � � 2 √ − 1 but not periodic. In ( [7], Remark 1), we have recently observed that any Bloch (ω, k)-periodic function must be almost periodic (see also the research articles [8] by Hasler and [9] by Hasler and Guérékata, where it has been noted that the Bloch (ω, k)-periodic functions are unavoidable in condensed matter and solid state physics). e notion of almost periodicity was introduced by Harald Bohr, a younger brother of Nobel Prize winner Niels Bohr, around 1925 and later generalized by many other mathematicians. In [10], we have analyzed the following generalization of the notion of almost periodicity, called c-almost periodicity (c ∈ C∖ 0 { }): let f: I ⟶ E be a continuous function, and let a number ϵ > 0 be given. We call a number τ > 0 an (ϵ, c)-period for f(·) if and only if ‖f(x + τ) − cf(x)‖ ≤ ϵ for all x ∈ I; by ϑ c (f, ϵ) we denote the set consisting of all (ϵ, c)-periods for f(·). It is said that f(·) is c-almost periodic if and only if for each ϵ > 0 the set ϑ c (f, ϵ) is relatively dense in [0, ∞), which means that for each ϵ > 0 there exists a finite real number l > 0 such that any subinterval I ′ of [0, ∞) of length l meets ϑ c (f, ϵ). Any c-periodic function is c-almost periodic and any c-almost periodic function is almost periodic ( [10]); if c � 1, resp. c � − 1, then we also say that the function f(·) is almost periodic, resp. almost antiperiodic (for the primary source of information about almost periodic functions and their applications, we refer the reader to the research monographs by Besicovitch [11], Diagana [12], Fink [13], Guérékata [14], Kostić [15], and Zaidman [16]).
In [10], besides the class of c-almost periodic functions, we have introduced and analyzed the classes of c-uniformly recurrent functions, semi-c-periodic functions, and their Stepanov generalizations, where c ∈ C and |c| � 1 (the classes of semiperiodic functions and semi-antiperiodic functions, i.e., the classes of semi-1-periodic functions and semi-(− 1)-periodic functions, have been previously considered by Andres and Pennequin in [17], the research article of invaluable importance for us, and Chaouchi et al. in [7]; the notion of semi-Bloch k-periodicity, where k ∈ R, has been also analyzed in [7], but it differs from the notion of semi-c-periodicity analyzed in [10] and this paper). If |c| � 1, then we know that a function f ∈ C(I: E) is semi-c-periodic if and only if there exists a sequence (f n ) of c-periodic functions in C(I: E) such that lim n⟶∞ f n (x) � f(x) uniformly in I; in this case, a semi-c-periodic function need not be c-periodic [10]. For example, we have the following (see ([17], Example 1), ( [7], Example 4 and Example 5), and ( [10], Example 2.16)): let p and q be odd natural numbers such that p − 1 ≡ 0(mod q), and let c � e (iπp/q) . e function is semi-c-periodic because it is a uniform limit of Our main result, eorem 1, states that the following phenomenon occurs in case |c| ≠ 1: if (f n ) is a sequence of c-periodic functions and lim n⟶∞ f n (x) � f(x) uniformly in I, then f(·) is c-periodic. erefore, in this case, any concept of semi-c-periodicity introduced below coincides with the concept of c-periodicity (more precisely, in this paper, we analyze the concepts of semi-c-periodicity of type i (i + ), where i � 1, 2 and c ∈ C∖ 0 { }; if |c| � 1, all these concepts are equivalent and reduced to the concept of semi-c-periodicity, while in case |c| ≠ 1, all these concepts are equivalent and reduced to the concept of c-periodicity).
For any function f ∈ C(I: e notion of c-uniform recurrence plays an important role in the proof of our main result [10]. if and only if there exists a strictly increasing sequence (α n ) of positive real numbers such that lim n⟶+∞ α n � +∞ and e space consisting of all c-uniformly recurrent functions from the interval I into E will be denoted by UR c (I: E). If c � 1, resp. c � − 1, then we also say that the function f(·) is uniformly recurrent, resp. uniformly antirecurrent.
Although the notion of uniform recurrence was analyzed already by Bohr in his landmark paper [18] (1924), the precise definition of a uniformly recurrent function was firstly given by Haraux and Souplet [19] in 2004, who proved that the function f: R ⟶ R, given by is unbounded, Lipschitz continuous and uniformly recurrent; moreover, we have that f(·) is c-uniformly recurrent if and only if c � 1 (see [10], Example 2.19(i)). e first example of a uniformly antirecurrent function has recently been constructed in ( [10], Example 2.20), where we have proved that the function g: R ⟶ R, given by is unbounded, Lipschitz continuous and uniformly antirecurrent. Any c-almost periodic function is c-uniformly recurrent, while the converse statement does not hold in general. For completeness, we will include all details of the proof of the following auxiliary lemma from [10].

Semi-c-Periodic Functions
Set S ≔ N if I � [0, ∞), and S ≔ Z if I � R. In this paper, we introduce and analyze the following notion with c ∈ C∖ 0 { }.

Definition 2. Let f ∈ C(I: E).
(i) It is said that f(·) is semi-c-periodic of type 1 if and only if (ii) It is said that f(·) is semi-c-periodic of type 2 if and only if e space of all semi-c-periodic functions of type i will be denoted by SP c,i (I: E), i � 1, 2.

Definition 3. Let f ∈ C(I: E).
(i) It is said that f(·) is semi-c-periodic of type 1 + if and only if (ii) It is said that f(·) is semi-c-periodic of type 2 + if and only if e space of all semi-c-periodic functions of type i + will be denoted by SP c,i,+ (I: E), i � 1, 2.
e notion of semi-c-periodicity of type 1 has been introduced in ([10], Definition 2.4), where it has been simply called semi-c-periodicity. Due to ([10], Proposition 2.5), we have that the notion of a semi-c-periodicity of type i (i + ), where i � 1, 2, is equivalent with the notion of semi-c-periodicity introduced there, provided that |c| � 1. Now we will focus our attention to the general case c ∈ C∖ 0 { }. We will first state the following.
□ e argumentation contained in the proofs of ([17], Lemma 1 and eorem 1) can be repeated verbatim in order to see that the following important lemma holds true. Proof. Suppose that the function f(·) is (ω, c)-periodic. en, we have f(x + mω) � c m f(x), x ∈ I, m ∈ S, so that f(·) is automatically semi-c-periodic of type i (i + ). To prove the converse statement, let us observe that any semi-c-periodic of type i is clearly semi-c-periodic of type i + . Suppose first that |c| > 1. Due to Lemma 2 B(i), it suffices to show that if f(·) is semi-c-periodic of type 2 + , then f(·) is c-periodic. Assume first I � [0, ∞). Using Lemma C, we get the existence of a sequence (f n : (0, ∞) ⟶ E) n∈N of c-periodic functions which converges uniformly to f(·). Let f n (x + ω n ) � cf n (x), x ≥ 0 for some sequence (ω n ) of positive real numbers. Consider first case that (ω n ) is bounded. en, there exists a strictly increasing sequence (n k ) of positive integers and a number ω ≥ 0 such that lim k⟶+∞ ω n k � ω. Let ϵ > 0 be given. en, there exists an integer k 0 ∈ N such that ‖f(x) − f n k (x)‖ ≤ ϵ/(2 + 2|c| − 1 ) for all real numbers x ≥ 0 and all integers k ≥ k 0 . Furthermore, we have for all real numbers x ≥ 0 and all integers k ≥ k 0 . Letting en, with the same notation as above, we may assume that lim k⟶+∞ ω n k � +∞. Using the same computation, it follows that lim k⟶+∞ ‖c − 1 f(· + ω n k ) − f(·)‖ ∞ � 0, so that f ∈ UR c ([0, ∞): E). Due to Lemma 1 A, we get f(·) ≡ 0. Assume now I � R. By the foregoing arguments, we know that there exists ω > 0 such that f(x + ω) � cf(x) for all x ≥ 0. Let x < 0 and ϵ > 0 be fixed. Since f(·) is semi-c-periodic, there exists ω ϵ > 0 such that ‖c − m f(x + ω + mω ϵ ) − f(x + ω)‖ ≤ ϵ and ‖c 1− m f(x + mω ϵ )− cf(x)‖ ≤ ϵ for all m ∈ N. For all sufficiently large integers m ∈ N, we have x + mω ϵ > 0 so that c − m f(x + ω + mω ϵ ) � c 1− m f(x + mω ϵ ), and therefore ‖f(x + ω) − cf(x)‖ ≤ 2ϵ. Since ϵ > 0 was arbitrary, we get f(x + ω) � cf(x), which completes the proof in case |c| > 1. Suppose now that |c| < 1. Due to Lemma 2(ii), it suffices to show that if f(·) is semi-c-periodic of type 1 + , then f(·) is c-periodic. But, then we can apply Lemma 3 again and the similar arguments as above to complete the whole proof. Using ( [10], eorem 2.14) and the proof of eorem 1, we may deduce the following corollaries.

Conclusions
In this paper, the authors have studied the class of semic-periodic functions with values in Banach spaces. In the case that c is a nonzero complex number whose absolute value is not equal to 1, the authors have proved that the notion of semi-c-periodicity is equivalent with the notion of c-periodicity. For further information concerning Stepanov semi-c-periodic functions, composition principles for (Stepanov) semi-c-periodic functions, and related applications to the abstract semilinear Volterra integrodifferential equations in Banach spaces, the reader may consult the forthcoming research monograph [20].
Data Availability e data that support the findings of this study are available at https://www.researchgate.net/publication/342068071_SEMI-c-PERIODIC_FUNCTIONS_AND_APPLICATIONS (an extended version of the paper).

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.