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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.36 no.3 Antofagasta set. 2017

http://dx.doi.org/10.4067/S0716-09172017000300449 

Articles

Existence of positive periodic solutions for delay dynamic equations

Faycal Bouchelaghem1 

Abdelouaheb Ardjouni2 

Ahcene Djoudi3 

1 University of Annaba, Faculty of Sciences, Department of Mathematics, Algeria, e-mail: fyb1500@gmail.com

2 University of Souk Ahras, Department of Mathematics and Informatics, Algeria, e-mail: abd_ardjouni@yahoo.fr

3 University of Annaba, Faculty of Sciences, Applied Mathematics Lab., Algeria, e-mail: adjoudi@yahoo.com

Abstract

In this article we study the existence of positive periodic solutions for a dynamic equations on time scales. The main tool employed here is the Schauder's fixed point theorem. The results obtained here extend the work of Olach (12). Two examples are also given to illustrate this work.

Keywords : Positive periodic solutions; Schauder's fixed point theorem; dynamic equations; time scales.

1.Introduction

In 1988, Stephan Hilger (8) introduced the theory of time scales (measure chains) as a means of unifying discrete and continuum calculi. Since Hilger's initial work there has been significant growth in the theory of dynamic equations on time scales, covering a variety of different problems; see (6,7,11) and references therein.

Let T be a periodic time scale such that t0 T. In this article, we are interested in the analysis of qualitative theory of positive periodic solutions of delay dynamic equations. Motivated by the papers (1)-(5, 9, 10, 12) and the references therein, we consider the following delay dynamic equation

Throughout this paper we assume that p,q: t 0, ∞) ∩ TR are rd-continuous, τ: TT is increasing so that the function x(τ(t)) is well defined over T. We also assume that τ: t 0, ∞) ∩ T 0, ∞) ∩ T is rd-continuous, τ(t) < t and limt→∞ τ(t) = ∞.To reach our desired end we have to transform (1.1) into an integral equation and then use Schauder's fixed point theorem to show the existence of positive periodic solutions.

The organization of this paper is as follows. In Section 2, we introduce some notations and definitions, and state some preliminary material needed in later sections. We will state some facts about the exponential function on a time scale as well as the Schauder's fixed point theorem. For details on Schauder theorem we refer the reader to (13). In Section 3, we establish our main results for positive periodic solutions by applying the Schauder's fixed point theorem. In Section 4, we present two examples to illustrate our results. The results presented in this paper extend the main results in (12)

2.Preliminaries

A time scale is an arbitrary nonempty closed subset of real numbers. The study of dynamic equations on time scales is a fairly new subject, and research in this area is rapidly growing (1-5, 9, 10, 12 and papers therein). The theory of dynamic equations unifies the theories of differential equations and difference equations. We suppose that the reader is familiar with the basic concepts concerning the calculus on time scales for dynamic equations. Otherwise one can find in Bohner and Peterson books (6, 7, 11) most of the material needed to read this paper. We start by giving some definitions necessary for our work. The notion of periodic time scales is introduced in Kaufmann and Raffoul (9). The following two definitions are borrowed from (9).

Definition 2.1. We say that a time scale T is periodic if there exist a p > 0 such that if tT then t ± pT. For TR, the smallest positive p is called the period of the time scale.

Example 2.2. The following time scales are periodic.

Remark 2.3 (9) All periodic time scales are unbounded above and below.

Definition 2.4. Let TR be a periodic time scale with period p. We say that the function f: TR is periodic with period ω if there exists a natural number n such that ω = np, f (t ± ω) = f (t) for all tT and ω is the smallest number such that f (t ± ω) = f(t).

If T = R, we say that f is periodic with period ω > 0 if ω is the smallest positive number such that f (t ± ω) = f (t) for all tT.

Remark 2.5 (9) If T is a periodic time scale with period p, then σ(t ± np) = σ(t) ± np. Consequently, the graininess function μ satisfies μ(t ± np) = σ(t ± np) - (t ± np) = σ(t) - t = μ (t) and so,is a periodic function with period p.

Definition 2.6.(6) A function f: TR is called rd-continuous provided it is continuous at every right-dense point tT and its left-sided limits exist, and is finite at every left-dense point tT. The set of rd-continuous functions f: TR will be denoted by

Crd = Crd (T) = Crd (T,R).

Definition 2.7. (6) For f: TR, we define f (t) to be the number (if it exists) with the property that for any given ε > 0, there exists a neighborhood U of t such that

|(f(σ(t)) - f(s)) - f (t)(σ(t)- s)| < ε|σ(t)-s| for all s ∈ U.

The function f : T kR is called the delta (or Hilger) derivative of f on T k.

Definition 2.8. (6)A function p: TR is called regressive provided 1+ μ(t)p(t) ≠ 0 for all tT. The set of all regressive and rd-continuous functions p: TR will be denoted by R = R (T,R). We define the set R+ of all positively regressive elements of R by

Definition 2.9. (6) Let pR, then the generalized exponential function ep is defined as the unique solution of the initial value problem

where log is the principal logarithm function.

Lemma 2.10 (6). Let p,q ∈ R. Then

Lemma 2.11 (1). If p ∈ R+, then

The proof of the main results in the next section is based upon an application of the following Schauder's fixed point theorem.

Theorem 2.12 (Schauder's fixed point theorem (13)) Let ω be a closed, convex and nonempty subset of a Banach space X. Let S: Ω→Ω be a continuous mapping such that SΩ is a relatively compact subset of X. Then S has at least one fixed point in ω. That is there exists an x ∈ Ω such that Sx = x.

3.The existence of periodic solutions

In this the section we will study existence of positive ω-periodic solution of Eq. (1.1). In the next lemma and theorem we choose tT sufficiently large that τ(t) ≥ t0 for tT.

Lemma 3.1. Suppose that there exists a rd-continuous function k: T, ∞) ∩ T → (0, ∞)such that

Then the function

Theorem 3.2.Suppose that there exists a rd-continuous function K: T, ∞) ∩ T → (0, ∞)such that (3.1) holds and

Then Eq. (1.1) has a positive ω-periodic solution.

Proof. Let X = C rd ( t 0, ∞) ∩ T,R) be the Banach space with the norm||x|| = sup t ≥ t0 |x(t)|. With regard to Lemma 3.1 we define

We now define a closed, bounded and convex subset ω of X as follows

Define the operator S:Ω → X as follows

We will show that for any x ∈ Ω we have Sx ∈ Ω. For every x ∈ Ω and tT we get

Furthermore for x ∈ Ω and tT we obtain

For t ∈ t0, T T we have ( Sx )(t) = 1, that is (S x)(t) ∈ Ω.

Further for every x ∈ Ω and τ(t) ≥ T we get

With regard to (3.2) and (3) for τ(t) ≥ T it follows that

Finally we will show that for x ∈ Ω, tT the function Sx is ω-periodic. For x ∈ Ω, tT and according to (3.1) we have

So Sx is ω- periodic on T, ∞) ∩ T. Thus we have proved that Sx ∈ Ω for any x ∈ Ω.

We now show that S is completely continuous. Let xi ∈ Ω be such that xi x ∈ Ω as i → ∞. For tT, we have

By applying the Lebesgue dominated convergence theorem we obtain that

For t ∈ t 0, T T the relation above is also valid. This means that S is continuous.

We now show that SΩ is relatively compact. It is sufficient to show by the Arzela-Ascoli theorem that the family of function {Sx: x ∈ Ω} is uniformly bounded and equicontinuous on t 0, ∞) ∩ T. The uniform boundedness follows from the definition of ω. With regard to (3.3) for tT, x ∈ Ω we get

This shows the equicontinuity of the family SΩ. Hence SΩ is relatively compact and therefore S is completely continuous. By Theorem 2.12 there is an x 0 ∈ Ω such that Sx 0 = x 0. We see that x 0 is a positive ω-periodic solution of Eq. (1.1). The proof is complete.

4.Two examples

In this section, we give two examples to illustrate the applications of Theorem 3.2.

Example 4.1.

Consider the delay dynamic equation on T with μ(t) ≠ 0,

We take k(t) = 1. Then for conditions (3.1), (3.2) and ω = 2π we obtain

All conditions of Theorem 3.2 are satisfied. Thus Eq. (4.1) has a positive ω = 2π- periodic solution

Example 4.2. Consider the delay differential equation on T = R,

We choose k(t) = e cos t . Then for conditions (3.1), (3.2) and ω = 2π we have

All conditions of Theorem 3.2 are satisfied. Thus Eq. (4.2) has a positive ω = 2π -periodic solution

for tT.

References

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[2] A. Ardjouni and A. Djoudi, Existence of positive periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale, Malaya Journal of Matematik 2 (1), pp. 60-67, (2013). [ Links ]

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Received: September 2016; Accepted: August 2017

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