Elsevier

Applied Mathematics and Computation

Volume 273, 15 January 2016, Pages 190-200
Applied Mathematics and Computation

On quasi-periodic properties of fractional sums and fractional differences of periodic functions

https://doi.org/10.1016/j.amc.2015.09.082Get rights and content

Abstract

This article is devoted to the study of discrete fractional calculus; our goal is to investigate quasi-periodic properties of fractional order sums and differences of periodic functions. Using Riemann–Liouville and Caputo type definitions, we study concepts close to the well known idea of periodic function, such as asymptotically periodicity or S-asymptotically periodicity. We use basic tools of discrete fractional calculus. Boundedness of sums and differences of fractional order of a given periodic function is also investigated.

Introduction

The ideas of fractional calculus, that is, integrals and derivatives of non integer order, have a long tradition. The birth of this theory comes back to Leibniz’s note in his letter to L’Hôpital dated 30 September 1695.

The key which motivated the most used definition of fractional order integral of a given function is the Cauchy formula for n-multiple integrals. Then, using this definition, the fractional order derivative of a function is introduced. These two concepts are the starting point of the theory of differential equations of non integer order, which has received increasing attention during recent years since fractional order differential operators provide an excellent instrument to study memory processes and properties of some materials. For a detailed exposition of fractional calculus, we refer the reader to references [13], [16], [17].

However, the theory of fractional difference calculus, which involves sums and differences of fractional order, is much less developed. In [14], Miller and Ross initiated the study of discrete fractional calculus. Some notions of fractional order differences can be found in [1], while essentials of fractional difference equations are the subject matter of some recent papers [4], [5], [6], [7], [8].

Periodic functions play a major role in mathematics. Indeed, the study of existence of periodic solutions and oscillatory behaviors is one of the most interesting and important topics in qualitative theory of differential equations. Of course, this is due to its implications in pure and applied analysis.

Nevertheless, the definition of periodic function is extremely demanding and then, conditions to guarantee the existence of periodic solutions are sometimes very harsh. For this reason, in the past decades, many authors (see [10], [11], [15] and references therein) have proposed and studied extensions of the well known idea of periodic function which have shown to be really interesting and useful in some situations.

It is an obvious fact that the classical derivative, if it exists, of a periodic function is also a periodic function with the same period. Moreover, the primitive of a periodic function may be periodic. The same holds also for discrete difference and sum operators. Nevertheless, when we consider derivatives or integrals of non integer order, this is not true (see, for example, [2], [18]). In [3] , we have studied some properties of fractional integrals and derivatives of periodic functions.

Motivated by references [3], [8], we present here the analogous results in the field of discrete fractional calculus. In the next section we introduce some definitions and notations. In Section 3 we prove that the fractional sum or difference of a periodic function is not a periodic function. Moreover, we consider S-asymptotically periodicity properties of discrete fractional operators. In Section 4 asymptotically periodicity properties are studied. Section 5 deals with boundedness of fractional sums and differences and finally, in Section 6 we point out one important difference between continuum and discrete fractional operators.

Section snippets

Some basic definitions and results

Throughout the paper, we assume the property of empty sum; that is, if a > b then t=abf(t)=0.Given tR, we set Nt={t,t+1,t+2,}.

Let us recall the definitions of difference and sum of fractional order, which have been previously introduced in [14].

Definition 2.1

Let α > 0, the fractional sum of f of order α with base point aZ, is defined by Δaαf(t)=1Γ(α)s=atα(ts1)(α1)f(s),where f is a function defined for s=amod1 and the falling factorial power function is defined, for αR, as t(α)={Γ(t+1)Γ(t+1α),ift+

Results about periodicity and S-asymptotically periodicity

In [8, Theorem 3.1], authors have proved that the fractional sum of non integer order of a non-zero N-periodic function is not an N-periodic function. However, since in next sections we will study quasi-periodic properties, we must prove that such fractional sum is not a periodic function for any integer period.

As we show in the next theorem, proof of this fact follows in a similar way as in the cited case.

Theorem 3.1

Let α > 0, αN, and f a non zero N-periodic function defined onNa. ThenΔaαf is not anN˜

Results about asymptotically periodicity

In the next theorem we will suppose that, given a periodic function f, Δaαf is a bounded function.

Theorem 4.1

Let f an N-periodic function withNN such that N > 1. IfΔaαf, with 0 < α < 1, is a bounded function, thenΔaαf is an asymptotically N-periodic function.

Proof

Let us denote for brevity φ(t)=Δaαf(t)=1Γ(α)s=atα(ts1)(α1)f(s),tNa+α.

For each nN, we consider the following functions φn(t)=φ(t+nN) and Φn(t)=supknφk(t) for tNa+α. Since φ is a bounded function, φn and Φn are also bounded functions

Results about boundedness

To know when the fractional sum of order 0 < α < 1 of a periodic function is a bounded function seems to be an important question. Hereinafter we study this question.

First of all we prove some results about Ra-transform introduced in [4].

Theorem 5.1 Final Value Theorem

Let f be a function defined inNa. Then, lims0+sRa[f(t)](s)=limn+f(a+n)if right-hand side limit exists.

Proof

We have that, Ra[Δf(t)](s)=t=a(1s+1)t+1Δf(t)=t=a(1s+1)t+1(f(t+1)f(t))=t=a(1s+1)t+1f(t+1)t=a(1s+1)t+1f(t)=t=a+1(1s+1)tf(t)t=a(1s+1)t+1f(t)=(s

Approximation of fractional integrals by fractional sums

Considering the h-calculus instead of the conventional difference calculus, we can use fractional sums to approximate fractional integrals. We refer to [6], for the details of h-calculus.

It is known that the fractional integral, with point base 0, of f(t)=cost of order 1 < α < 2 is a bounded function (see [3, Section 5]).

However, if we consider fractional order sums of order 1 < α < 2, we have that hΔ0αcos(t)=hΔ0(α1)hΔ01cos(t),t{hα,2hα,3hα,},where (see [12, Theorem 2.5]) hΔ01cos(t)=sin(t

Acknowledgments

We would like to thank the two anonymous referees for their many helpful and detailed comments.

The work of I. Area has been partially supported by the Ministerio de Economía y Competitividad of Spain under grant MTM2012–38794–C02–01, co-financed by the European Community fund FEDER. J.J. Nieto also acknowledges partial financial support by the Ministerio de Economía y Competitividad of Spain under grant MTM2013–43014–P co-financed by the European Community fund FEDER. J. Losada acknowledges

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