# On Semi-c-Periodic Functions

Authors: M. T. Khalladi; M. Kostić; M. Pinto; A. Rahmani; D. Velinov
Journal: Journal of Mathematics
 (2021)

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## Abstract

The main aim of this paper is to indicate that the notion of semi-c-periodicity is equivalent with the notion of c-periodicity, provided that c is a nonzero complex number whose absolute value is not equal to 1.


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## Body

## 1. Introduction

The notion of periodicity plays a fundamental role in mathematics. A continuous functionf: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 fx+ω=fx for all x∈I. The 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 fx+ω=cfx 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−⋅/ωf⋅ is periodic and gx+ω=gx 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, byE we denote a complex Banach space equipped with the norm ‖⋅‖; CI:E denotes the vector space consisting of all continuous functions f:I⟶E. A function f∈CI:E is said to be c-periodic (c∈ℂ∖0) if and only if there exists a real number ω>0 such that the function f⋅ is ω,c-periodic. The class of c-periodic functions extends two important classes of functions:(1)
The class of antiperiodic functions, i.e., the class of−1-periodic functions: in this case, any positive real number ω>0 satisfying fx+ω=−fx, 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 that fx+2ω=−fx for all x∈I.(2)
The class of Blochω,k-periodic functions (ω>0, k∈ℝ), i.e., the class of continuous functions f:I⟶E satisfying fx+ω=eikωfx for all x∈I. The 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 fx+mω=eimkωfx 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(1)fx≔eix+ei2−1x,x∈ℝ,is Blochω,k-periodic with ω=2π+2π 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).The 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 fx+τ−cfx≤ϵ for all x∈I; by ϑcf,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 ϑcf,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 ϑcf,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∈CI:E is semi-c-periodic if and only if there exists a sequence fn of c-periodic functions in CI:E such that limn⟶∞fnx=fx 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≡0modq, and let c=eiπp/q. The function(2)fx≔∑n=1∞eix/2nq+1n2,x∈ℝ,is semi-c-periodic because it is a uniform limit of π⋅1+2q…1+2Nq-periodic functions(3)fNx≔∑n=1Neix/2nq+1n2,x∈ℝN∈ℕ.Our main result, Theorem1, states that the following phenomenon occurs in case c≠1: if fn is a sequence of c-periodic functions and limn⟶∞fnx=fx uniformly in I, then f⋅ is c-periodic. Therefore, 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 functionf∈CI:E, we set f∞≔supx∈Ifx. The notion of c-uniform recurrence plays an important role in the proof of our main result [10].Definition 1.
A continuous functionf:I⟶E is said to be c-uniformly recurrent (c∈ℂ∖0) if and only if there exists a strictly increasing sequence αn of positive real numbers such that limn⟶+∞αn=+∞ and(4)limn⟶+∞f⋅+αn−cf⋅∞=0.
The space consisting of allc-uniformly recurrent functions from the interval I into E will be denoted by URcI: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(5)fx≔∑n=1∞1nsin2x2n,x∈ℝ,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)). The 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(6)gx≔sinx⋅∑n=1∞1nsin2x3n,x∈ℝ,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].Lemma 1.
(A). Suppose thatf∈URcI:E and c∈ℂ∖0 satisfies c≠1. Then, f≡0.Proof.
Without loss of generality, we may assume thatI=0,∞. Suppose to the contrary that there exists x0≥0 such that fx0≠0. Inductively, (4) implies(7)ckm−ck−1nc−1≤fx≤ckM−ck−1nc−1,provided that k∈ℕ and x∈kαn,k+1αn. Consider now case c<1. Let 0<ϵ<cfx0. Then, (7) yields that there exist integers k0∈ℕ and n∈ℕ such that for each k∈ℕ with k≥k0, we have fx≤ϵ/2, x∈kαn,k+1αn. Then, the contradiction is obvious because for each m∈ℕ with m>n, there exists k∈ℕ such that x0+αm∈kαn,k+1αn, and therefore fx0+αm≥cfx0−1/m⟶cfx0>ϵ, m⟶+∞. Consider now case c>1; let n∈ℕ be such that fx0>1/nc−1 and M≔maxx∈0,2αnfx>0. Then, for each m∈ℕ with m>n, there exists k∈ℕ such that αm∈k−1αn,kαn, and therefore fx+αm≤1+cM, x∈0,2αn. On the other hand, we obtain inductively from (4) that(8)fx0+kαn≥ckfx0−1nc−1+1nc−1⟶+∞ask∈ℕ,which immediately yields a contradiction.


## 2. Semi-c-Periodic Functions

SetS≔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.
Letf∈CI:E.(i)
It is said thatf⋅ is semi-c-periodic of type 1 if and only if(9)∀ε>0∃ω>0∀m∈S∀x∈Ifx+mω−cmfx≤ε.(ii)
It is said thatf⋅ is semi-c-periodic of type 2 if and only if(10)∀ε>0∃ω>0∀m∈S∀x∈Ic−mfx+mω−fx≤ε.
The space of all semi-c-periodic functions of type i will be denoted by SPc,iI:E, i=1,2.Definition 3.
Letf∈CI:E.(i)
It is said thatf⋅ is semi-c-periodic of type 1+ if and only if(11)∀ε>0∃ω>0∀m∈ℕ∀x∈Ifx+mω−cmfx≤ε.(ii)
It is said thatf⋅ is semi-c-periodic of type 2+ if and only if(12)∀ε>0∃ω>0∀m∈ℕ∀x∈Ic−mfx+mω−fx≤ε.
The space of all semi-c-periodic functions of type i+ will be denoted by SPc,i,+I:E, i=1,2.The 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 casec∈ℂ∖0. We will first state the following.Lemma 2.
(B).(i)
Ifc≥1 and f:I⟶E is semi-c-periodic of type 1+, then f⋅ is semi-c-periodic of type 2+.(ii)
Ifc≤1 and f:I⟶E is semi-c-periodic of type 2+, then f⋅ is semi-c-periodic of type 1+.Proof.
Ifx∈I, ω>0, m∈ℕ and c≥1, then we have(13)fx+mω−cmfx≤ε⇒c−mfx+mω−fx≤ε,which implies (i); the proof of (ii) is similar.The argumentation contained in the proofs of ([17], Lemma 1 and Theorem 1) can be repeated verbatim in order to see that the following important lemma holds true.Lemma 3.
(C). Suppose thatc≤1, resp. c≥1, and f:0,∞⟶E is semi-c-periodic of type 1+, resp. 2+. Then, there exists a sequencefn:0,∞⟶En∈ℕofc-periodic functions which converges uniformly tof⋅.Now we are able to state and prove our main result.Theorem 1.
Letc≠1, i∈1,2 and f:I⟶E. Then, f⋅ is c-periodic if and only if f⋅ is semi-c-periodic of typei(i+).Proof.
Suppose that the functionf⋅ is ω,c-periodic. Then, we have fx+mω=cmfx, 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 fn:0,∞⟶En∈ℕ of c-periodic functions which converges uniformly to f⋅. Let fnx+ωn=cfnx, x≥0 for some sequence ωn of positive real numbers. Consider first case that ωn is bounded. Then, there exists a strictly increasing sequence nk of positive integers and a number ω≥0 such that limk⟶+∞ωnk=ω. Let ϵ>0 be given. Then, there exists an integer k0∈ℕ such that fx−fnkx≤ϵ/2+2c−1 for all real numbers x≥0 and all integers k≥k0. Furthermore, we have(14)c−1fx+ωnk−fx≤c−1fx+ωnk−c−1fnkx+ωnk+c−1fnkx+ωnk−fnkx+fnkx−fx=c−1fx+ωnk−c−1fnkx+ωnk+fnkx−fx≤21+c−1ϵ2+2c−1=ϵ,for all real numbers x≥0 and all integers k≥k0. Letting k⟶+∞, we get fx+ω=cfx for all x≥0. If ω>0, the above yields that f⋅ is ω,c-periodic while the assumption ω=0 yields f≡0 or c=1, i.e., f⋅≡0; in any case, f⋅ is ω,c-periodic. Suppose now that ωn is unbounded. Then, with the same notation as above, we may assume that limk⟶+∞ωnk=+∞. Using the same computation, it follows that limk⟶+∞c−1f⋅+ωnk−f⋅∞=0, so that f∈URc0,∞: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 fx+ω=cfx for all x≥0. Let x<0 and ϵ>0 be fixed. Since f⋅ is semi-c-periodic, there exists ωϵ>0 such that c−mfx+ω+mωϵ−fx+ω≤ϵ and c1−mfx+mωϵ−cfx≤ϵ for all m∈ℕ. For all sufficiently large integers m∈ℕ, we have x+mωϵ>0 so that c−mfx+ω+mωϵ=c1−mfx+mωϵ, and therefore fx+ω−cfx≤2ϵ. Since ϵ>0 was arbitrary, we get fx+ω=cfx, 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.Corollary 1.
Letc∈ℂ∖0, let i∈1,2, and let f⋅ be semi-c-periodic of type i (i+). Then, there exist two finite real constantsM>0andω>0such thatfx≤Mcx/ω, t∈I.Using ([10], Theorem 2.14) and the proof of Theorem 1, we may deduce the following corollaries.Corollary 2.
Letf∈CI:E and c∈ℂ∖0. Then, f⋅ is semi-c-periodic if and only if there exists a sequence fnofc-periodic functions inCI:Esuch thatlimn⟶∞fnx=fxuniformly inI.Corollary 3.
Letf∈CI:E and |c|≠1. If fn is a sequence of c-periodic functions and limn⟶∞fnx=fxuniformly inI, thenf⋅isc-periodic.


## 3. Conclusions

In this paper, the authors have studied the class of semi-c-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].



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