A new periodicity concept for time scales

Abstract

ABSTRACT. By means of the shift operators we introduce a new periodicity concept on time scales. This new approach will enable researchers to investigate periodicity notion on a large class of time scales whose members may not satisfy the condition there exists a P > 0 such that t ± P ∈ T for all t ∈ T, which is being currently used. Therefore, the results of this paper open an avenue for the investigation of periodic solutions of q-difference equations and more.

Multiple positive periodic solutions in shifts &delta;<SUB align="right">± for an impulsive nabla dynamic equation on time scales

“…Definition 1: Adivar (2013), periodicity in shifts δ ± : Let T be a time scale with the shift operators δ ± associated with the initial point * 0…”

“…T Definition 2: Adivar (2013), periodic function in shifts δ ± : Let T be a time scale that is periodic in shifts δ ± with the period . P A real-value function f defined on * T is periodic in shifts δ ± if there exists…”

Periodicity, almost periodicity for time scales and related functions

“…The rst concepts of periodic time scales and periodic functions were proposed by Kaufmann and Ra oul in [7], and they were improved by Adıvar in [9]. However, the concepts of periodic time scales proposed by [7][8][9] were without considering the shift direction of time scales, which will be quite limited for understanding the recent results from [2,10] etc. For an example, consider…”

“…In fact, for any t ∈ T, although δ+( , t) = t ∈ T, there exists t = ∈ T such that δ−( , t ) = t / = / ̸ ∈ T. However, (1.2) also has very nice closedness in shift for time variables if the time scale is only shifted towards the positive direction, because for any t ∈ T, we have δ+( , t) ∈ T. Hence, concepts of periodic time scales from [7,9] were without considering the shift direction of time scales. In this paper we propose a new concept of periodic time scales with "translation direction" so that we can regard (1.1) as a periodic time scale.…”

“…In fact, for any t ∈ T, we have t + ∈ T, but there exists t = ∈ T such that t − = − ̸ ∈ T. Nevertheless, (1.1) has very nice closedness in translation for time variables if the time scale is only translated towards the positive direction, because for any t ∈ T, we have t + ∈ T. For another example, consider 2) according to the new concept of periodic time scales from [9], (1.2) is also not a periodic time scales since T has a in mum that equals to with respect to the shift operator δ−. In fact, for any t ∈ T, although δ+( , t) = t ∈ T, there exists t = ∈ T such that δ−( , t ) = t / = / ̸ ∈ T. However, (1.2) also has very nice closedness in shift for time variables if the time scale is only shifted towards the positive direction, because for any t ∈ T, we have δ+( , t) ∈ T. Hence, concepts of periodic time scales from [7,9] were without considering the shift direction of time scales.…”