Periodicity on isolated time scales

Böhner, Martin; Mesquita, Jaqueline G.; Streipert, Sabrina

doi:10.1002/mana.201900360

Cited by 7 publications

(10 citation statements)

References 28 publications

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“…We generalize () to arbitrary isolated time scales in Lemma 2.2 and will see that it simplifies the study of delay dynamic equations significantly. It also provides the functional structure of periodic functions, see Theorem 2.3, extending a result in Bohner et al 17 to higher periods.…”

Section: Periodic Functions On Time Scalessupporting

confidence: 55%

“…By Bohner et al, 17, Theorem 4.9 we have for

t, t_{0} \in 𝕋_{scriptI}

and

p \in {scriptP}_{ω}

regressive,

e_{p} false(t, ρ^{ω} false(t false) false) = e_{p} false(t_{0}, ρ^{ω} false(t_{0} false) false) .

…”

Section: Periodic Functions On Time Scalesmentioning

confidence: 99%

“…Equation ( 5) can be used to identify the structure of periodic functions, generalizing Theorem 5.1 in Bohner et al, 17 which classified one-periodic functions as…”

Section: 𝜇(T)mentioning

confidence: 99%

“…In Bohner et al, 17, Lemma 4.6 the authors show that for all

ω \in ℕ_{1} = false{1,2, \dots false}, {scriptP}_{ω} \subset {scriptP}_{2 ω}

, that is, all ω ‐periodic functions are 2 ω ‐periodic. We generalize this result in the proceeding lemma.…”

Section: Periodic Functions On Time Scalesmentioning

confidence: 99%

“…Popular examples of such isolated time scales are the discrete domain

ℤ

and the quantum time scales

q^{ℕ_{0}} = false{1, q, q^{2}, \dots false}

with q > 1. The restriction to isolated time scales allows the applications of the recently introduced definition of periodicity in Bohner et al 17 and avoids the restrictive, but commonly applied, assumption of periodic time scales, see Wang et al 18 and the references therein. In the special case of

𝕋 = ℤ

, a function is ω ‐periodic if f ( n + ω ) = f ( n ) for all

n \in ℤ

.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Delay dynamic equations on isolated time scales and the relevance of one‐periodic coefficients

Böhner

Cuchta

Streipert

2022

Math Methods in App Sciences

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We are motivated by the idea that certain properties of delay differential and difference equations with constant coefficients arise as a consequence of their one-periodic nature. We apply the recently introduced definition of periodicity for arbitrary isolated time scales to linear delay dynamic equations and a class of nonlinear delay dynamic equations. Utilizing a derived identity of higher order delta derivatives and delay terms, we rewrite the considered linear and nonlinear delayed dynamic equations with one-periodic coefficients as a linear autonomous dynamic system with constant matrix. As the simplification of a constant matrix is only obtained for one-periodic coefficients, dynamic equations with one-periodic coefficients are the simplest form compared to the commonly used constant coefficients.

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Section: Periodic Functions On Time Scalessupporting

confidence: 55%

“…By Bohner et al, 17, Theorem 4.9 we have for

t, t_{0} \in 𝕋_{scriptI}

and

p \in {scriptP}_{ω}

regressive,

e_{p} false(t, ρ^{ω} false(t false) false) = e_{p} false(t_{0}, ρ^{ω} false(t_{0} false) false) .

…”

Section: Periodic Functions On Time Scalesmentioning

confidence: 99%

“…Equation ( 5) can be used to identify the structure of periodic functions, generalizing Theorem 5.1 in Bohner et al, 17 which classified one-periodic functions as…”

Section: 𝜇(T)mentioning

confidence: 99%

“…In Bohner et al, 17, Lemma 4.6 the authors show that for all

ω \in ℕ_{1} = false{1,2, \dots false}, {scriptP}_{ω} \subset {scriptP}_{2 ω}

, that is, all ω ‐periodic functions are 2 ω ‐periodic. We generalize this result in the proceeding lemma.…”

Section: Periodic Functions On Time Scalesmentioning

confidence: 99%

“…Popular examples of such isolated time scales are the discrete domain

ℤ

and the quantum time scales

q^{ℕ_{0}} = false{1, q, q^{2}, \dots false}

𝕋 = ℤ

, a function is ω ‐periodic if f ( n + ω ) = f ( n ) for all

n \in ℤ

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Delay dynamic equations on isolated time scales and the relevance of one‐periodic coefficients

Böhner

Cuchta

Streipert

2022

Math Methods in App Sciences

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Generalized periodicity and applications to logistic growth

Bohner,

Mesquita,

Streipert

2024

Chaos, Solitons & Fractals

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Permanence of equilibrium points in the basin of attraction and existence of periodic solutions for autonomous measure differential equations and dynamic equations on time scales via generalized ODEs

et al. 2022

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It is well-known that generalized ODEs encompass several types of differential equations as, for instance, functional differential equations, measure differential equations, dynamic equations on time scales, impulsive differential equations and any combinations among them, not to mention integrals equations, among others. The aim of this paper is to establish a theory of autonomous equations in the setting of generalized ODEs. Thus, we introduce the notion of autonomous generalized ODEs as well as new classes of right-hand sides for nonautonomous generalized ODEs. Amongst the main results, we prove that one of these new classes coincide with the original class of right-hand sides introduced by Kurzweil in 1957. We also prove that autonomous generalized ODEs do not enlarge the class of autonomous ODEs with uniformly continuous right-hand sides. Motivated by this fact, we then enlarge the class of autonomous generalized ODEs so that discontinuities can be taken into account. We then introduce a more general class of autonomous generalized ODEs, in whose integral form, a Stieltjes-type integral appears. A correspondence between these equations and autonomous measure differential equations is established and several results are obtained. We mention local existence and uniqueness of solutions, continuous dependence of solutions on initial values, existence of periodic solutions and permanence of asymptotically stable equilibrium point in the basin of attraction. All these results are, then, specified not only for autonomous generalized ODEs, but also for autonomous measure differential equations and dynamic equations on time scales.

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Periodicity on isolated time scales

Cited by 7 publications

References 28 publications

Delay dynamic equations on isolated time scales and the relevance of one‐periodic coefficients

Delay dynamic equations on isolated time scales and the relevance of one‐periodic coefficients

Generalized periodicity and applications to logistic growth

Permanence of equilibrium points in the basin of attraction and existence of periodic solutions for autonomous measure differential equations and dynamic equations on time scales via generalized ODEs

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