Research ArticleOn Semi-c-Periodic Functions

M T Khalladi 1 M Kostic 2 M Pinto 3 A Rahmani4 and D Velinov 5

1Department of Mathematics and Computer Sciences University of Adrar Adrar Algeria2Faculty of Technical Sciences University of Novi Sad Trg D Obradovica 6 21125 Novi Sad Serbia3Departamento de Matematicas Facultad de Ciencias Universidad de Chile Santiago de Chile Chile4Laboratory of Mathematics Modeling and Applications (LaMMA) University of Adrar Adrar Algeria5Faculty of Civil Engineering Ss Cyril and Methodius University Skopje Partizanski Odredi 24 PO Box 560 1000 SkopjeNorth Macedonia

Correspondence should be addressed to M Kostic marcosveratnet

Received 26 November 2020 Revised 17 January 2021 Accepted 19 January 2021 Published 31 January 2021

Academic Editor Yongqiang Fu

Copyright copy 2021M T Khalladi et alis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

e notion of periodicity plays a fundamental role inmathematics A continuous function f I⟶ E where E isa topological space and I R or I [0infin) is said to beperiodic if and only if there exists a real number ωgt 0 suchthat f(x + ω) f(x) for all x isin I e notion of periodicityhas recently been reconsidered by Alvarez et al [1] whoproposed the following notion a continuous functionf I⟶ E where E is a complex Banach space is said to be(ω c)-periodic (ωgt 0 c isin C∖ 0 ) if and only if f(x + ω)

cf(x) for all x isin I Due to ([1] Proposition 22) we knowthat a continuous function f I⟶ E is (ω c)-periodic ifand only if the function g(middot) equiv c(minus middotω)f(middot) is periodic andg(x + ω) g(x) for all x isin I here c(minus middotω) denotes theprincipal branch of the exponential function (see also theresearch articles [2 3] by Alvarez et al the conferencepaper [4] by Pinto where the idea for introduction of(ω c)-periodic functions was presented for the first timeand [5 6] for some generalizations of the concept of(ω c)-periodicity)

In the sequel by E we denote a complex Banach spaceequipped with the norm middot C(I E) denotes the vectorspace consisting of all continuous functions f I⟶ E Afunction f isin C(I E) is said to be c-periodic (c isin C∖ 0 ) if

and only if there exists a real number ωgt 0 such that thefunction f(middot) is (ω c)-periodic e class of c-periodicfunctions extends two important classes of functions

(1) e class of antiperiodic functions ie the class of(minus 1)-periodic functions in this case any positivereal number ωgt 0 satisfying f(x + ω) minus f(x)x isin I is said to be an antiperiod of f(middot) Anyantiperiodic function is periodic since we can applythe above functional equality twice in order to seethat f(x + 2ω) minus f(x) for all x isin I

(2) e class of Bloch (ω k)-periodic functions (ωgt 0k isin R) ie the class of continuous functionsf I⟶ E satisfying f(x + ω) eikωf(x) for allx isin I e number ω is usually called Bloch periodof f(middot) the number k is usually called the Blochwave vector or Floquet exponent of f(middot) and in thecase that kω π the class of Bloch (ω k)-periodicfunctions is equal to the class of antiperiodicfunctions having the number ω as an antiperiod Ifthe function f(middot) is Bloch (ω k)-periodic then weinductively obtain f(x + mω) eimkωf(x) for allx isin I and m isin N so that the function f(middot) must beperiodic provided that kω isin Q but if kω notin Q thenthe function f(middot) need not be periodic as the

HindawiJournal of MathematicsVolume 2021 Article ID 6620625 5 pageshttpsdoiorg10115520216620625

following simple counterexample shows thefunction

f(x) ≔ eix

+ ei(

2

radicminus 1)x

x isin R (1)

is Bloch (ω k)-periodic with ω 2π +2

radicπ and k

2radic

minus 1 but not periodic In ([7] Remark 1) we haverecently observed that any Bloch (ω k)-periodicfunction must be almost periodic (see also the re-search articles [8] by Hasler and [9] by Hasler andGuerekata where it has been noted that the Bloch(ω k)-periodic functions are unavoidable in con-densed matter and solid state physics)

e notion of almost periodicity was introduced byHarald Bohr a younger brother of Nobel Prize winner NielsBohr around 1925 and later generalized by many othermathematicians In [10] we have analyzed the followinggeneralization of the notion of almost periodicity calledc-almost periodicity (c isin C∖ 0 ) let f I⟶ E be a con-tinuous function and let a number ϵgt 0 be given We call anumber τ gt 0 an (ϵ c)-period for f(middot) if and only if f(x +

τ) minus cf(x) le ϵ for all x isin I by ϑc(f ϵ) we denote the setconsisting of all (ϵ c)-periods for f(middot) It is said that f(middot) isc-almost periodic if and only if for each ϵgt 0 the set ϑc(f ϵ)is relatively dense in [0infin) which means that for each ϵgt 0there exists a finite real number lgt 0 such that any subin-terval Iprime of [0infin) of length l meets ϑc(f ϵ) Any c-periodicfunction is c-almost periodic and any c-almost periodicfunction is almost periodic ([10]) if c 1 resp c minus 1 thenwe also say that the function f(middot) is almost periodic respalmost antiperiodic (for the primary source of informationabout almost periodic functions and their applications werefer the reader to the research monographs by Besicovitch[11] Diagana [12] Fink [13] Guerekata [14] Kostic [15]and Zaidman [16])

In [10] besides the class of c-almost periodic functionswe have introduced and analyzed the classes of c-uniformlyrecurrent functions semi-c-periodic functions and theirStepanov generalizations where c isin C and |c| 1 (theclasses of semiperiodic functions and semi-antiperiodicfunctions ie the classes of semi-1-periodic functions andsemi-(minus 1)-periodic functions have been previously con-sidered by Andres and Pennequin in [17] the research articleof invaluable importance for us and Chaouchi et al in [7]the notion of semi-Bloch k-periodicity where k isin R hasbeen also analyzed in [7] but it differs from the notion ofsemi-c-periodicity analyzed in [10] and this paper) If |c| 1then we know that a function f isin C(I E) is semi-c-periodicif and only if there exists a sequence (fn) of c-periodicfunctions in C(I E) such that limn⟶infinfn(x) f(x)

uniformly in I in this case a semi-c-periodic function neednot be c-periodic [10] For example we have the following(see ([17] Example 1) ([7] Example 4 and Example 5) and([10] Example 216)) let p and q be odd natural numberssuch that p minus 1 equiv 0(mod q) and let c e(iπpq) e function

f(x) ≔ 1113944

infin

n1

e(ix(2nq+1))

n2 x isin R (2)

is semi-c-periodic because it is a uniform limit of[π middot (1 + 2q) (1 + 2Nq)]-periodic functions

fN(x) ≔ 1113944N

n1

e(ix(2nq+1))

n2 x isin R (N isin N) (3)

Our main result eorem 1 states that the followingphenomenon occurs in case |c|ne 1 if (fn) is a sequence ofc-periodic functions and limn⟶infinfn(x) f(x) uniformly inI then f(middot) is c-periodic erefore in this case any conceptof semi-c-periodicity introduced below coincides with theconcept of c-periodicity (more precisely in this paper weanalyze the concepts of semi-c-periodicity of type i (i+) wherei 1 2 and c isin C∖ 0 if |c| 1 all these concepts areequivalent and reduced to the concept of semi-c-periodicitywhile in case |c|ne 1 all these concepts are equivalent andreduced to the concept of c-periodicity)

For any function f isin C(I E) we set finfin ≔ supxisinI

f(x) e notion of c-uniform recurrence plays an im-portant role in the proof of our main result [10]

Definition 1 A continuous function f I⟶ E is said to bec-uniformly recurrent (c isin C∖ 0 ) if and only if there exists astrictly increasing sequence (αn) of positive real numberssuch that limn⟶+infinαn +infin and

limn⟶+infin

f middot + αn( 1113857 minus cf(middot)

infin 0 (4)

e space consisting of all c-uniformly recurrent func-tions from the interval I into E will be denoted by URc(I E)If c 1 resp c minus 1 then we also say that the function f(middot)

is uniformly recurrent resp uniformly antirecurrent

Although the notion of uniform recurrence was analyzedalready by Bohr in his landmark paper [18] (1924) theprecise definition of a uniformly recurrent function wasfirstly given by Haraux and Souplet [19] in 2004 who provedthat the function f R⟶ R given by

f(x) ≔ 1113944infin

n1

1nsin2

x

2n1113874 1113875 x isin R (5)

is unbounded Lipschitz continuous and uniformly recur-rent moreover we have that f(middot) is c-uniformly recurrent ifand only if c 1 (see [10] Example 219(i)) e first ex-ample of a uniformly antirecurrent function has recentlybeen constructed in ([10] Example 220) where we haveproved that the function g R⟶ R given by

g(x) ≔ (sinx) middot 1113944infin

n1

1nsin2

x

3n1113874 1113875 x isin R (6)

is unbounded Lipschitz continuous and uniformly anti-recurrent Any c-almost periodic function is c-uniformlyrecurrent while the converse statement does not hold ingeneral

For completeness we will include all details of the proofof the following auxiliary lemma from [10]

2 Journal of Mathematics

Lemma 1 (A) Suppose that f isin URc(I E) and c isin C∖ 0

satisfies |c|ne 1 en f equiv 0

Proof Without loss of generality we may assume thatI [0infin) Suppose to the contrary that there exists x0 ge 0such that f(x0)ne 0 Inductively (4) implies

|c|km minus

|c|k

minus 1n(|c| minus 1)

le f(x) le |c|kM minus

|c|k

minus 1n(|c| minus 1)

(7)

provided that k isin N and x isin [kαn (k + 1)αn] Consider nowcase |c|lt 1 Let 0lt ϵlt cf(x0) en (7) yields that thereexist integers k0 isin N and n isin N such that for each k isin N withkge k0 we have f(x)le (ϵ2) x isin [kαn (k + 1)αn] enthe contradiction is obvious because for each m isin N withmgt n there exists k isin N such that x0 + αm isin [kαn

(k + 1)αn] and therefore f(x0 + αm)ge |c|f(x0)minus

(1m)⟶ |c|f(x0)gt ϵ m⟶ +infin Consider now case|c|gt 1 let n isin N be such that f(x0)gt (1(n(|c| minus 1))) andM ≔ maxxisin[02αn]f(x) gt 0 en for each m isin N withmgt n there exists k isin N such that αm isin [(k minus 1)αn kαn] andtherefore f(x + αm)le 1 + |c|M x isin [0 2αn] On the otherhand we obtain inductively from (4) that

f x0 + kαn( 1113857

ge |c|k

f x0( 1113857

minus1

n(|c| minus 1)1113890 1113891

+1

n(|c| minus 1)⟶ +infin as k isin N

(8)

which immediately yields a contradiction

2 Semi-c-Periodic Functions

Set S ≔ N if I [0infin) and S ≔ Z if I R In this paperwe introduce and analyze the following notion withc isin C∖ 0

Definition 2 Let f isin C(I E)

(i) It is said that f(middot) is semi-c-periodic of type 1 if andonly if

forallεgt 0existωgt 0forallm isin Sforallx isin I f(x + mω) minus cm

f(x)

le ε

(9)

(ii) It is said that f(middot) is semi-c-periodic of type 2 if andonly if

forallεgt 0existωgt 0forallm isin Sforallx isin I cminus m

f(x + mω) minus f(x)

le ε

(10)

e space of all semi-c-periodic functions of type i will bedenoted by SPci(I E) i 1 2

Definition 3 Let f isin C(I E)

(i) It is said that f(middot) is semi-c-periodic of type 1+ if andonly if

forallεgt 0existωgt 0forallm isin Nforallx isin I f(x + mω) minus cm

f(x)

le ε

(11)

(ii) It is said that f(middot) is semi-c-periodic of type 2+ if andonly if

forallεgt 0existωgt 0forallm isin Nforallx isin I cminus m

f(x + mω) minus f(x)

le ε

(12)

e space of all semi-c-periodic functions of type i+ willbe denoted by SPci+(I E) i 1 2

e notion of semi-c-periodicity of type 1 has beenintroduced in ([10] Definition 24) where it has been simplycalled semi-c-periodicity Due to ([10] Proposition 25) wehave that the notion of a semi-c-periodicity of type i (i+)where i 1 2 is equivalent with the notion of semi-c-pe-riodicity introduced there provided that |c| 1

Now we will focus our attention to the general casec isin C∖ 0 We will first state the following

Lemma 2 (B)

(i) If |c|ge 1 and f I⟶ E is semi-c-periodic of type 1+then f(middot) is semi-c-periodic of type 2+

(ii) If |c|le 1 and f I⟶ E is semi-c-periodic of type 2+then f(middot) is semi-c-periodic of type 1+

Proof If x isin I ωgt 0 m isin N and |c|ge 1 then we have

f(x + mω) minus cm

f(x)

le εrArr cminus m

f(x + mω) minus f(x)

le ε(13)

which implies (i) the proof of (ii) is similar

e argumentation contained in the proofs of ([17]Lemma 1 andeorem 1) can be repeated verbatim in orderto see that the following important lemma holds true

Lemma 3 (C) Suppose that |c|le 1 resp |c|ge 1 andf [0infin)⟶ E is semi-c-periodic of type 1+ resp 2+ enthere exists a sequence (fn [0infin)⟶ E)nisinN of c-periodicfunctions which converges uniformly to f(middot)

Now we are able to state and prove our main result

Theorem 1 Let |c|ne 1 i isin 1 2 and f I⟶ E en f(middot)

is c-periodic if and only if f(middot) is semi-c-periodic of type i (i+)

Proof Suppose that the functionf(middot) is (ω c)-periodicenwe have f(x + mω) cmf(x) x isin I m isin S so that f(middot) isautomatically semi-c-periodic of type i (i+) To prove theconverse statement let us observe that any semi-c-periodic oftype i is clearly semi-c-periodic of type i+ Suppose first that|c|gt 1 Due to Lemma 2 B(i) it suffices to show that if f(middot) issemi-c-periodic of type 2+ thenf(middot) is c-periodic Assume firstI [0infin) Using LemmaC we get the existence of a sequence(fn (0infin)⟶ E)nisinN of c-periodic functions which

Journal of Mathematics 3

converges uniformly to f(middot) Let fn(x + ωn) cfn(x) xge 0for some sequence (ωn) of positive real numbers Consider firstcase that (ωn) is bounded en there exists a strictly in-creasing sequence (nk) of positive integers and a number ωge 0such that limk⟶+infinωnk

ω Let ϵgt 0 be given en thereexists an integer k0 isin N such that f(x) minus fnk

(x)le ϵ(2 +

2|c|minus 1) for all real numbers xge 0 and all integers kge k0Furthermore we have

cminus 1

f x + ωnk1113872 1113873 minus f(x)

le cminus 1

f x + ωnk1113872 1113873 minus c

minus 1fnk

x + ωnk1113872 1113873

+ cminus 1

fnkx + ωnk

1113872 1113873 minus fnk(x)

+ fnk(x) minus f(x)

cminus 1

f x + ωnk1113872 1113873 minus c

minus 1fnk

x + ωnk1113872 1113873

+ fnk(x) minus f(x)

le 2 1 +|c|minus 1

1113872 1113873

middotϵ

2 + 2|c|minus 1

1113872 1113873 ϵ

(14)

for all real numbers xge 0 and all integers kge k0 Lettingk⟶ +infin we get f(x + ω) cf(x) for all xge 0 If ωgt 0the above yields that f(middot) is (ω c)-periodic while the as-sumptionω 0 yieldsf equiv 0 or c 1 ief(middot) equiv 0 in any casef(middot) is (ω c)-periodic Suppose now that (ωn) is unboundeden with the same notation as above we may assume thatlimk⟶+infinωnk

+infin Using the same computation it followsthat limk⟶+infincminus 1f(middot + ωnk

) minus f(middot)infin 0 so thatf isin URc([0infin) E) Due to Lemma 1 A we get f(middot) equiv 0Assume now I R By the foregoing arguments we know thatthere exists ωgt 0 such that f(x + ω) cf(x) for all xge 0 Letxlt 0 and ϵgt 0 be fixed Since f(middot) is semi-c-periodic thereexists ωϵ gt 0 such that cminus mf(x + ω + mωϵ) minus f(x + ω)le ϵand c1minus mf(x + mωϵ)minus cf(x)le ϵ for all m isin N For allsufficiently large integers m isin N we have x + mωϵ gt 0 so thatcminus mf(x + ω + mωϵ) c1minus mf(x + mωϵ) and thereforef(x + ω) minus cf(x)le 2ϵ Since ϵgt 0 was arbitrary we getf(x + ω) cf(x) which completes the proof in case |c|gt 1Suppose now that |c|lt 1 Due to Lemma 2(ii) it suffices to showthat iff(middot) is semi-c-periodic of type 1+ thenf(middot) is c-periodicBut then we can apply Lemma 3 again and the similar argu-ments as above to complete the whole proof

Corollary 1 Let c isin C∖ 0 let i isin 1 2 and let f(middot) besemi-c-periodic of type i (i+) en there exist two finite realconstantsMgt 0 and ωgt 0 such that f(x)leM|c|(xω) t isin I

Using ([10] eorem 214) and the proof of eorem 1we may deduce the following corollaries

Corollary 2 Let f isin C(I E) and c isin C∖ 0 en f(middot) issemi-c-periodic if and only if there exists a sequence (fn) ofc-periodic functions in C(I E) such that limn⟶infinfn(x)

f(x) uniformly in I

Corollary 3 Let f isin C(I E) and |c|ne 1 If (fn) is a se-quence of c-periodic functions and limn⟶infinfn(x) f(x)

uniformly in I then f(middot) is c-periodic

3 Conclusions

In this paper the authors have studied the class of semi-c-periodic functions with values in Banach spaces In thecase that c is a nonzero complex number whose absolutevalue is not equal to 1 the authors have proved that thenotion of semi-c-periodicity is equivalent with the notion ofc-periodicity For further information concerning Stepanovsemi-c-periodic functions composition principles for(Stepanov) semi-c-periodic functions and related applica-tions to the abstract semilinear Volterra integrodifferentialequations in Banach spaces the reader may consult theforthcoming research monograph [20]

Data Availability

e data that support the findings of this study are available athttpswwwresearchgatenetpublication342068071_SEMI-c-PERIODIC_FUNCTIONS_AND_APPLICATIONS (an ex-tended version of the paper)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Marko Kostic was partially supported by the Ministry ofScience and Technological Development Republic of Serbia(grant no 451-03-68202014200156) Manuel Pinto waspartially supported by FONDECYT (1170466)

References

[1] E Alvarez A Gomez andM Pinto ldquo(ω c)-periodic functionsand mild solutions to abstract fractional integro-differentialequationsrdquo Electronic Journal of Qualitative eory of Dif-ferential Equations vol 16 no 16 pp 1ndash8 2018

[2] E Alvarez S Castillo and M Pinto ldquo(ω c)-pseudo periodicfunctions first order Cauchy problem and Lasota-Wazewskamodel with ergodic and unbounded oscillating production ofred cellsrdquo Boundary Value Problems vol 2019 no 1 pp 1ndash202019

[3] E Alvarez S Castillo and M Pinto ldquo(ω c)-asymptoticallyperiodic functions first-order Cauchy problem andLasota-Wazewska model with unbounded oscillatingproduction of red cellsrdquo Mathematical Methods in theApplied Sciences vol 43 no 1 pp 305ndash319 2020

[4] M Pinto Ergodicity and Oscillations Presented at the Con-ference in Universidad Catolica del Norte Antofagasta Chile2014

[5] M T Khalladi M Kostic A Rahmani and D Velinov(ω c)-Almost Periodic Type Functions and Applicationspreprint hal-02549066f 2020

[6] M T Khalladi M Kostic A Rahmani and D Velinovldquo(ω c)-Pseudo almost periodic functions (ω c)- pseudoalmost automorphic functions and applicationsrdquo Facta

4 Journal of Mathematics

Universitatis Series Mathematics and Informatics inpress 2020

[7] B Chaouchi M Kostic S Pilipovic and D VelinovldquoSemi-Bloch periodic functions semi-anti-periodicfunctions and applicationsrdquo Chelj Physical MathematicalJournal vol 5 no 2 pp 243ndash255 2020

[8] M F Hasler ldquoBloch-periodic generalized functionsrdquo NoviSad Journal of Mathematics vol 46 no 2 pp 135ndash143 2016

[9] M F Hasler and G M Nrsquo Guerekata ldquoBloch-periodicfunctions and some applicationsrdquo Nonlinear Studies vol 21no 1 pp 21ndash30 2014

[10] M T Khalladi M Kostic M Pinto A Rahmani andD Velinov ldquo-Almost periodic functions and applicationsrdquoNonautonomous Dynamic Systems vol 7 pp 176ndash193 2020

[11] A S Besicovitch Almost Periodic Functions Dover NewYork NY USA 1954

[12] T Diagana Almost Automorphic Type and Almost PeriodicType Functions in Abstract Spaces Springer New York NYUSA 2013

[13] A M Fink Almost Periodic Differential Equations SpringerBerlin Germany 1974

[14] G M Nrsquo Guerekata Almost Automorphic and Almost Peri-odic Functions in Abstract Spaces Kluwer Dordrecht eNetherlands 2001

[15] M Kostic Almost Periodic and Almost Automorphic TypeSolutions to Integro-Differential Equations W de GruyterBerlin Germany 2019

[16] S Zaidman ldquoAlmost-periodic functions in abstract spacesrdquoPitman Research Notes in Math Vol 126 Pitman Boston1985

[17] J Andres and D Pennequin ldquoSemi-periodic solutions ofdifference and differential equationsrdquo Boundary ValueProblems vol 2012 no 1 pp 1ndash16 2012

[18] B Bohr ldquoZur theorie der fastperiodischen Funktionen IrdquoActa Math vol 45 pp 29ndash127 1924

[19] A Haraux and P Souplet ldquoAn example of uniformly re-current function which is not almost periodicrdquo Journal ofFourier Analysis and Applications vol 10 no 2 pp 217ndash2202004

[20] M Kostic Selected Topics in Almost Periodicity Book-manuscript 2020

Journal of Mathematics 5

Lemma 1 (A) Suppose that f isin URc(I E) and c isin C∖ 0

satisfies |c|ne 1 en f equiv 0

Proof Without loss of generality we may assume thatI [0infin) Suppose to the contrary that there exists x0 ge 0such that f(x0)ne 0 Inductively (4) implies

|c|km minus

|c|k

minus 1n(|c| minus 1)

le f(x) le |c|kM minus

|c|k

minus 1n(|c| minus 1)

(7)

provided that k isin N and x isin [kαn (k + 1)αn] Consider nowcase |c|lt 1 Let 0lt ϵlt cf(x0) en (7) yields that thereexist integers k0 isin N and n isin N such that for each k isin N withkge k0 we have f(x)le (ϵ2) x isin [kαn (k + 1)αn] enthe contradiction is obvious because for each m isin N withmgt n there exists k isin N such that x0 + αm isin [kαn

(k + 1)αn] and therefore f(x0 + αm)ge |c|f(x0)minus

(1m)⟶ |c|f(x0)gt ϵ m⟶ +infin Consider now case|c|gt 1 let n isin N be such that f(x0)gt (1(n(|c| minus 1))) andM ≔ maxxisin[02αn]f(x) gt 0 en for each m isin N withmgt n there exists k isin N such that αm isin [(k minus 1)αn kαn] andtherefore f(x + αm)le 1 + |c|M x isin [0 2αn] On the otherhand we obtain inductively from (4) that

f x0 + kαn( 1113857

ge |c|k

f x0( 1113857

minus1

n(|c| minus 1)1113890 1113891

+1

n(|c| minus 1)⟶ +infin as k isin N

(8)

which immediately yields a contradiction

2 Semi-c-Periodic Functions

Set S ≔ N if I [0infin) and S ≔ Z if I R In this paperwe introduce and analyze the following notion withc isin C∖ 0

Definition 2 Let f isin C(I E)

(i) It is said that f(middot) is semi-c-periodic of type 1 if andonly if

forallεgt 0existωgt 0forallm isin Sforallx isin I f(x + mω) minus cm

f(x)

le ε

(9)

(ii) It is said that f(middot) is semi-c-periodic of type 2 if andonly if

forallεgt 0existωgt 0forallm isin Sforallx isin I cminus m

f(x + mω) minus f(x)

le ε

(10)

e space of all semi-c-periodic functions of type i will bedenoted by SPci(I E) i 1 2

Definition 3 Let f isin C(I E)

(i) It is said that f(middot) is semi-c-periodic of type 1+ if andonly if

forallεgt 0existωgt 0forallm isin Nforallx isin I f(x + mω) minus cm

f(x)

le ε

(11)

(ii) It is said that f(middot) is semi-c-periodic of type 2+ if andonly if

forallεgt 0existωgt 0forallm isin Nforallx isin I cminus m

f(x + mω) minus f(x)

le ε

(12)

e space of all semi-c-periodic functions of type i+ willbe denoted by SPci+(I E) i 1 2

e notion of semi-c-periodicity of type 1 has beenintroduced in ([10] Definition 24) where it has been simplycalled semi-c-periodicity Due to ([10] Proposition 25) wehave that the notion of a semi-c-periodicity of type i (i+)where i 1 2 is equivalent with the notion of semi-c-pe-riodicity introduced there provided that |c| 1

Now we will focus our attention to the general casec isin C∖ 0 We will first state the following

Lemma 2 (B)

(i) If |c|ge 1 and f I⟶ E is semi-c-periodic of type 1+then f(middot) is semi-c-periodic of type 2+

(ii) If |c|le 1 and f I⟶ E is semi-c-periodic of type 2+then f(middot) is semi-c-periodic of type 1+

Proof If x isin I ωgt 0 m isin N and |c|ge 1 then we have

f(x + mω) minus cm

f(x)

le εrArr cminus m

f(x + mω) minus f(x)

le ε(13)

which implies (i) the proof of (ii) is similar

e argumentation contained in the proofs of ([17]Lemma 1 andeorem 1) can be repeated verbatim in orderto see that the following important lemma holds true

Lemma 3 (C) Suppose that |c|le 1 resp |c|ge 1 andf [0infin)⟶ E is semi-c-periodic of type 1+ resp 2+ enthere exists a sequence (fn [0infin)⟶ E)nisinN of c-periodicfunctions which converges uniformly to f(middot)

Now we are able to state and prove our main result

Theorem 1 Let |c|ne 1 i isin 1 2 and f I⟶ E en f(middot)

is c-periodic if and only if f(middot) is semi-c-periodic of type i (i+)

Proof Suppose that the functionf(middot) is (ω c)-periodicenwe have f(x + mω) cmf(x) x isin I m isin S so that f(middot) isautomatically semi-c-periodic of type i (i+) To prove theconverse statement let us observe that any semi-c-periodic oftype i is clearly semi-c-periodic of type i+ Suppose first that|c|gt 1 Due to Lemma 2 B(i) it suffices to show that if f(middot) issemi-c-periodic of type 2+ thenf(middot) is c-periodic Assume firstI [0infin) Using LemmaC we get the existence of a sequence(fn (0infin)⟶ E)nisinN of c-periodic functions which

Journal of Mathematics 3

converges uniformly to f(middot) Let fn(x + ωn) cfn(x) xge 0for some sequence (ωn) of positive real numbers Consider firstcase that (ωn) is bounded en there exists a strictly in-creasing sequence (nk) of positive integers and a number ωge 0such that limk⟶+infinωnk

ω Let ϵgt 0 be given en thereexists an integer k0 isin N such that f(x) minus fnk

(x)le ϵ(2 +

2|c|minus 1) for all real numbers xge 0 and all integers kge k0Furthermore we have

cminus 1

f x + ωnk1113872 1113873 minus f(x)

le cminus 1

f x + ωnk1113872 1113873 minus c

minus 1fnk

x + ωnk1113872 1113873

+ cminus 1

fnkx + ωnk

1113872 1113873 minus fnk(x)

+ fnk(x) minus f(x)

cminus 1

f x + ωnk1113872 1113873 minus c

minus 1fnk

x + ωnk1113872 1113873

+ fnk(x) minus f(x)

le 2 1 +|c|minus 1

1113872 1113873

middotϵ

2 + 2|c|minus 1

1113872 1113873 ϵ

(14)

for all real numbers xge 0 and all integers kge k0 Lettingk⟶ +infin we get f(x + ω) cf(x) for all xge 0 If ωgt 0the above yields that f(middot) is (ω c)-periodic while the as-sumptionω 0 yieldsf equiv 0 or c 1 ief(middot) equiv 0 in any casef(middot) is (ω c)-periodic Suppose now that (ωn) is unboundeden with the same notation as above we may assume thatlimk⟶+infinωnk

+infin Using the same computation it followsthat limk⟶+infincminus 1f(middot + ωnk

) minus f(middot)infin 0 so thatf isin URc([0infin) E) Due to Lemma 1 A we get f(middot) equiv 0Assume now I R By the foregoing arguments we know thatthere exists ωgt 0 such that f(x + ω) cf(x) for all xge 0 Letxlt 0 and ϵgt 0 be fixed Since f(middot) is semi-c-periodic thereexists ωϵ gt 0 such that cminus mf(x + ω + mωϵ) minus f(x + ω)le ϵand c1minus mf(x + mωϵ)minus cf(x)le ϵ for all m isin N For allsufficiently large integers m isin N we have x + mωϵ gt 0 so thatcminus mf(x + ω + mωϵ) c1minus mf(x + mωϵ) and thereforef(x + ω) minus cf(x)le 2ϵ Since ϵgt 0 was arbitrary we getf(x + ω) cf(x) which completes the proof in case |c|gt 1Suppose now that |c|lt 1 Due to Lemma 2(ii) it suffices to showthat iff(middot) is semi-c-periodic of type 1+ thenf(middot) is c-periodicBut then we can apply Lemma 3 again and the similar argu-ments as above to complete the whole proof

Corollary 1 Let c isin C∖ 0 let i isin 1 2 and let f(middot) besemi-c-periodic of type i (i+) en there exist two finite realconstantsMgt 0 and ωgt 0 such that f(x)leM|c|(xω) t isin I

Using ([10] eorem 214) and the proof of eorem 1we may deduce the following corollaries

Corollary 2 Let f isin C(I E) and c isin C∖ 0 en f(middot) issemi-c-periodic if and only if there exists a sequence (fn) ofc-periodic functions in C(I E) such that limn⟶infinfn(x)

f(x) uniformly in I

Corollary 3 Let f isin C(I E) and |c|ne 1 If (fn) is a se-quence of c-periodic functions and limn⟶infinfn(x) f(x)

uniformly in I then f(middot) is c-periodic

3 Conclusions

In this paper the authors have studied the class of semi-c-periodic functions with values in Banach spaces In thecase that c is a nonzero complex number whose absolutevalue is not equal to 1 the authors have proved that thenotion of semi-c-periodicity is equivalent with the notion ofc-periodicity For further information concerning Stepanovsemi-c-periodic functions composition principles for(Stepanov) semi-c-periodic functions and related applica-tions to the abstract semilinear Volterra integrodifferentialequations in Banach spaces the reader may consult theforthcoming research monograph [20]

Data Availability

e data that support the findings of this study are available athttpswwwresearchgatenetpublication342068071_SEMI-c-PERIODIC_FUNCTIONS_AND_APPLICATIONS (an ex-tended version of the paper)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Marko Kostic was partially supported by the Ministry ofScience and Technological Development Republic of Serbia(grant no 451-03-68202014200156) Manuel Pinto waspartially supported by FONDECYT (1170466)

References

[1] E Alvarez A Gomez andM Pinto ldquo(ω c)-periodic functionsand mild solutions to abstract fractional integro-differentialequationsrdquo Electronic Journal of Qualitative eory of Dif-ferential Equations vol 16 no 16 pp 1ndash8 2018

[2] E Alvarez S Castillo and M Pinto ldquo(ω c)-pseudo periodicfunctions first order Cauchy problem and Lasota-Wazewskamodel with ergodic and unbounded oscillating production ofred cellsrdquo Boundary Value Problems vol 2019 no 1 pp 1ndash202019

[3] E Alvarez S Castillo and M Pinto ldquo(ω c)-asymptoticallyperiodic functions first-order Cauchy problem andLasota-Wazewska model with unbounded oscillatingproduction of red cellsrdquo Mathematical Methods in theApplied Sciences vol 43 no 1 pp 305ndash319 2020

[4] M Pinto Ergodicity and Oscillations Presented at the Con-ference in Universidad Catolica del Norte Antofagasta Chile2014

[5] M T Khalladi M Kostic A Rahmani and D Velinov(ω c)-Almost Periodic Type Functions and Applicationspreprint hal-02549066f 2020

[6] M T Khalladi M Kostic A Rahmani and D Velinovldquo(ω c)-Pseudo almost periodic functions (ω c)- pseudoalmost automorphic functions and applicationsrdquo Facta

4 Journal of Mathematics

Universitatis Series Mathematics and Informatics inpress 2020

[7] B Chaouchi M Kostic S Pilipovic and D VelinovldquoSemi-Bloch periodic functions semi-anti-periodicfunctions and applicationsrdquo Chelj Physical MathematicalJournal vol 5 no 2 pp 243ndash255 2020

[8] M F Hasler ldquoBloch-periodic generalized functionsrdquo NoviSad Journal of Mathematics vol 46 no 2 pp 135ndash143 2016

[9] M F Hasler and G M Nrsquo Guerekata ldquoBloch-periodicfunctions and some applicationsrdquo Nonlinear Studies vol 21no 1 pp 21ndash30 2014

[10] M T Khalladi M Kostic M Pinto A Rahmani andD Velinov ldquo-Almost periodic functions and applicationsrdquoNonautonomous Dynamic Systems vol 7 pp 176ndash193 2020

[11] A S Besicovitch Almost Periodic Functions Dover NewYork NY USA 1954

[12] T Diagana Almost Automorphic Type and Almost PeriodicType Functions in Abstract Spaces Springer New York NYUSA 2013

[13] A M Fink Almost Periodic Differential Equations SpringerBerlin Germany 1974

[14] G M Nrsquo Guerekata Almost Automorphic and Almost Peri-odic Functions in Abstract Spaces Kluwer Dordrecht eNetherlands 2001

[15] M Kostic Almost Periodic and Almost Automorphic TypeSolutions to Integro-Differential Equations W de GruyterBerlin Germany 2019

[16] S Zaidman ldquoAlmost-periodic functions in abstract spacesrdquoPitman Research Notes in Math Vol 126 Pitman Boston1985

[17] J Andres and D Pennequin ldquoSemi-periodic solutions ofdifference and differential equationsrdquo Boundary ValueProblems vol 2012 no 1 pp 1ndash16 2012

[18] B Bohr ldquoZur theorie der fastperiodischen Funktionen IrdquoActa Math vol 45 pp 29ndash127 1924

[19] A Haraux and P Souplet ldquoAn example of uniformly re-current function which is not almost periodicrdquo Journal ofFourier Analysis and Applications vol 10 no 2 pp 217ndash2202004

[20] M Kostic Selected Topics in Almost Periodicity Book-manuscript 2020

Journal of Mathematics 5

converges uniformly to f(middot) Let fn(x + ωn) cfn(x) xge 0for some sequence (ωn) of positive real numbers Consider firstcase that (ωn) is bounded en there exists a strictly in-creasing sequence (nk) of positive integers and a number ωge 0such that limk⟶+infinωnk

ω Let ϵgt 0 be given en thereexists an integer k0 isin N such that f(x) minus fnk

(x)le ϵ(2 +

2|c|minus 1) for all real numbers xge 0 and all integers kge k0Furthermore we have

cminus 1

f x + ωnk1113872 1113873 minus f(x)

le cminus 1

f x + ωnk1113872 1113873 minus c

minus 1fnk

x + ωnk1113872 1113873

+ cminus 1

fnkx + ωnk

1113872 1113873 minus fnk(x)

+ fnk(x) minus f(x)

cminus 1

f x + ωnk1113872 1113873 minus c

minus 1fnk

x + ωnk1113872 1113873

+ fnk(x) minus f(x)

le 2 1 +|c|minus 1

1113872 1113873

middotϵ

2 + 2|c|minus 1

1113872 1113873 ϵ

(14)

for all real numbers xge 0 and all integers kge k0 Lettingk⟶ +infin we get f(x + ω) cf(x) for all xge 0 If ωgt 0the above yields that f(middot) is (ω c)-periodic while the as-sumptionω 0 yieldsf equiv 0 or c 1 ief(middot) equiv 0 in any casef(middot) is (ω c)-periodic Suppose now that (ωn) is unboundeden with the same notation as above we may assume thatlimk⟶+infinωnk

+infin Using the same computation it followsthat limk⟶+infincminus 1f(middot + ωnk

) minus f(middot)infin 0 so thatf isin URc([0infin) E) Due to Lemma 1 A we get f(middot) equiv 0Assume now I R By the foregoing arguments we know thatthere exists ωgt 0 such that f(x + ω) cf(x) for all xge 0 Letxlt 0 and ϵgt 0 be fixed Since f(middot) is semi-c-periodic thereexists ωϵ gt 0 such that cminus mf(x + ω + mωϵ) minus f(x + ω)le ϵand c1minus mf(x + mωϵ)minus cf(x)le ϵ for all m isin N For allsufficiently large integers m isin N we have x + mωϵ gt 0 so thatcminus mf(x + ω + mωϵ) c1minus mf(x + mωϵ) and thereforef(x + ω) minus cf(x)le 2ϵ Since ϵgt 0 was arbitrary we getf(x + ω) cf(x) which completes the proof in case |c|gt 1Suppose now that |c|lt 1 Due to Lemma 2(ii) it suffices to showthat iff(middot) is semi-c-periodic of type 1+ thenf(middot) is c-periodicBut then we can apply Lemma 3 again and the similar argu-ments as above to complete the whole proof

Corollary 1 Let c isin C∖ 0 let i isin 1 2 and let f(middot) besemi-c-periodic of type i (i+) en there exist two finite realconstantsMgt 0 and ωgt 0 such that f(x)leM|c|(xω) t isin I

Using ([10] eorem 214) and the proof of eorem 1we may deduce the following corollaries

Corollary 2 Let f isin C(I E) and c isin C∖ 0 en f(middot) issemi-c-periodic if and only if there exists a sequence (fn) ofc-periodic functions in C(I E) such that limn⟶infinfn(x)

f(x) uniformly in I

Corollary 3 Let f isin C(I E) and |c|ne 1 If (fn) is a se-quence of c-periodic functions and limn⟶infinfn(x) f(x)

uniformly in I then f(middot) is c-periodic

3 Conclusions

In this paper the authors have studied the class of semi-c-periodic functions with values in Banach spaces In thecase that c is a nonzero complex number whose absolutevalue is not equal to 1 the authors have proved that thenotion of semi-c-periodicity is equivalent with the notion ofc-periodicity For further information concerning Stepanovsemi-c-periodic functions composition principles for(Stepanov) semi-c-periodic functions and related applica-tions to the abstract semilinear Volterra integrodifferentialequations in Banach spaces the reader may consult theforthcoming research monograph [20]

Data Availability

e data that support the findings of this study are available athttpswwwresearchgatenetpublication342068071_SEMI-c-PERIODIC_FUNCTIONS_AND_APPLICATIONS (an ex-tended version of the paper)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Marko Kostic was partially supported by the Ministry ofScience and Technological Development Republic of Serbia(grant no 451-03-68202014200156) Manuel Pinto waspartially supported by FONDECYT (1170466)

References

[1] E Alvarez A Gomez andM Pinto ldquo(ω c)-periodic functionsand mild solutions to abstract fractional integro-differentialequationsrdquo Electronic Journal of Qualitative eory of Dif-ferential Equations vol 16 no 16 pp 1ndash8 2018

[2] E Alvarez S Castillo and M Pinto ldquo(ω c)-pseudo periodicfunctions first order Cauchy problem and Lasota-Wazewskamodel with ergodic and unbounded oscillating production ofred cellsrdquo Boundary Value Problems vol 2019 no 1 pp 1ndash202019

[3] E Alvarez S Castillo and M Pinto ldquo(ω c)-asymptoticallyperiodic functions first-order Cauchy problem andLasota-Wazewska model with unbounded oscillatingproduction of red cellsrdquo Mathematical Methods in theApplied Sciences vol 43 no 1 pp 305ndash319 2020

[4] M Pinto Ergodicity and Oscillations Presented at the Con-ference in Universidad Catolica del Norte Antofagasta Chile2014

[5] M T Khalladi M Kostic A Rahmani and D Velinov(ω c)-Almost Periodic Type Functions and Applicationspreprint hal-02549066f 2020

[6] M T Khalladi M Kostic A Rahmani and D Velinovldquo(ω c)-Pseudo almost periodic functions (ω c)- pseudoalmost automorphic functions and applicationsrdquo Facta

4 Journal of Mathematics

Universitatis Series Mathematics and Informatics inpress 2020

[7] B Chaouchi M Kostic S Pilipovic and D VelinovldquoSemi-Bloch periodic functions semi-anti-periodicfunctions and applicationsrdquo Chelj Physical MathematicalJournal vol 5 no 2 pp 243ndash255 2020

[8] M F Hasler ldquoBloch-periodic generalized functionsrdquo NoviSad Journal of Mathematics vol 46 no 2 pp 135ndash143 2016

[9] M F Hasler and G M Nrsquo Guerekata ldquoBloch-periodicfunctions and some applicationsrdquo Nonlinear Studies vol 21no 1 pp 21ndash30 2014

[10] M T Khalladi M Kostic M Pinto A Rahmani andD Velinov ldquo-Almost periodic functions and applicationsrdquoNonautonomous Dynamic Systems vol 7 pp 176ndash193 2020

[11] A S Besicovitch Almost Periodic Functions Dover NewYork NY USA 1954

[12] T Diagana Almost Automorphic Type and Almost PeriodicType Functions in Abstract Spaces Springer New York NYUSA 2013

[13] A M Fink Almost Periodic Differential Equations SpringerBerlin Germany 1974

[14] G M Nrsquo Guerekata Almost Automorphic and Almost Peri-odic Functions in Abstract Spaces Kluwer Dordrecht eNetherlands 2001

[15] M Kostic Almost Periodic and Almost Automorphic TypeSolutions to Integro-Differential Equations W de GruyterBerlin Germany 2019

[16] S Zaidman ldquoAlmost-periodic functions in abstract spacesrdquoPitman Research Notes in Math Vol 126 Pitman Boston1985

[17] J Andres and D Pennequin ldquoSemi-periodic solutions ofdifference and differential equationsrdquo Boundary ValueProblems vol 2012 no 1 pp 1ndash16 2012

[18] B Bohr ldquoZur theorie der fastperiodischen Funktionen IrdquoActa Math vol 45 pp 29ndash127 1924

[19] A Haraux and P Souplet ldquoAn example of uniformly re-current function which is not almost periodicrdquo Journal ofFourier Analysis and Applications vol 10 no 2 pp 217ndash2202004

[20] M Kostic Selected Topics in Almost Periodicity Book-manuscript 2020

Journal of Mathematics 5

Universitatis Series Mathematics and Informatics inpress 2020

[7] B Chaouchi M Kostic S Pilipovic and D VelinovldquoSemi-Bloch periodic functions semi-anti-periodicfunctions and applicationsrdquo Chelj Physical MathematicalJournal vol 5 no 2 pp 243ndash255 2020

[8] M F Hasler ldquoBloch-periodic generalized functionsrdquo NoviSad Journal of Mathematics vol 46 no 2 pp 135ndash143 2016

[9] M F Hasler and G M Nrsquo Guerekata ldquoBloch-periodicfunctions and some applicationsrdquo Nonlinear Studies vol 21no 1 pp 21ndash30 2014

[10] M T Khalladi M Kostic M Pinto A Rahmani andD Velinov ldquo-Almost periodic functions and applicationsrdquoNonautonomous Dynamic Systems vol 7 pp 176ndash193 2020

[11] A S Besicovitch Almost Periodic Functions Dover NewYork NY USA 1954

[12] T Diagana Almost Automorphic Type and Almost PeriodicType Functions in Abstract Spaces Springer New York NYUSA 2013

[13] A M Fink Almost Periodic Differential Equations SpringerBerlin Germany 1974

[14] G M Nrsquo Guerekata Almost Automorphic and Almost Peri-odic Functions in Abstract Spaces Kluwer Dordrecht eNetherlands 2001

[15] M Kostic Almost Periodic and Almost Automorphic TypeSolutions to Integro-Differential Equations W de GruyterBerlin Germany 2019

[16] S Zaidman ldquoAlmost-periodic functions in abstract spacesrdquoPitman Research Notes in Math Vol 126 Pitman Boston1985

[17] J Andres and D Pennequin ldquoSemi-periodic solutions ofdifference and differential equationsrdquo Boundary ValueProblems vol 2012 no 1 pp 1ndash16 2012

[18] B Bohr ldquoZur theorie der fastperiodischen Funktionen IrdquoActa Math vol 45 pp 29ndash127 1924

[19] A Haraux and P Souplet ldquoAn example of uniformly re-current function which is not almost periodicrdquo Journal ofFourier Analysis and Applications vol 10 no 2 pp 217ndash2202004

[20] M Kostic Selected Topics in Almost Periodicity Book-manuscript 2020

Journal of Mathematics 5