Existence of periodic solutions of dynamic equations on time scales by averaging

In this paper, we study the existence of periodic solutions for perturbed dynamic equations on time scales. Our approach is based on the averaging method. Further, we extend some averaging theorem to periodic solutions of dynamic equations on time scales to $k-$th order in $\varepsilon$. More precisely, results of higher order averaging for finding periodic solutions are given via the topological degree theory.

1. Introduction and statement of main result. The problem of periodic solution of continuous dynamical system has been one of the center topics since Poincaré and Lyapunov. Recently, some theories and methods are developed to dynamic equations on time scales, see, for instance, [6,15,11,1,10]. However, the existence of periodic solutions for differential equations with a small parameter ε when 'time' is not continuous have paid more and more attention. The aim of this paper is to study the existence of periodic solutions for the perturbed dynamic equations on time scales of the type where f i : T × U → R n for i = 0, 1, · · · , k and r : T × U × (−ε 0 , ε 0 ) → R n are rd-continuous, and T -periodic in the first argument, U is an open subset of R n , and ε is a small parameter. A time scale is an arbitrary non-empty closed subset of R, generally denoted by T. The theory of time scales was first introduced by S. Hilger, on his doctoral thesis (see [9]) in order to study continuous-discrete hybrid processes. For instance, if T = Z, dynamic equations are just usual difference equations, while, taking T = R, they are usual differential equations. This theory respects a powerful tool to economics, populations models, biology models and so on. It has been attracting more and more attentions during the past years and the existence of solutions for systems on time scales has been extensively investigated, specially concerning periodicity (see, [1]). Periodic dynamic equations on time scales has been attracting attention of several mathematicians and the interest in this topic still increases. For instance, almost periodicity on time scales was introduced by Y. Li and C. Wang in [10]. The theory of almost automorphic solutions of dynamic equations on time scales was introduced by C. Lizama and J. G. Mesquita (see, [11]). After that, C. Wang and Y. Li (see, [15]) considered nonlinear dynamic equations and proved the existence of affine-periodic solutions via the topological degree theory.
A useful tool to study periodic solutions is the averaging theory. The method of averaging for ordinary differential equations has a long history that started with the classical works of Lagrange and Laplace on celestial mechanics. The first formalization of this procedure was given by Fatou in 1928 [8]. Bogoliubov [2] and Krylov [3] made important contributions to the averaging theorem. In 2004, Buica and Llibre [7] extended the averaging theory up to order 3 for studying periodic orbits of continuous differential systems by Brouwer's degree. Recently, higher order averaging theorem for finding periodic orbits of continuous differential systems has been introduced by Llibre et al. [12] and the theory of regularization was introduced by Llibre, Novaes and Teixeira in [13].
Motivated by these facts, the main goal of this paper is to study the existence of periodic solutions of the perturbed dynamic equations on time scales by the averaging method. The proof is inspired by the classical one, but certain technical details on time scales are more complicated. As well known, by using the implicit function theorem, the averaging method leads to the existence of periodic solutions for periodic systems(see, [14]). But in this paper, we use the topological method to prove our main results, which do not need smoothness.
In order to prove our main results, we introduce some notations. Let x = (x 1 , · · · , x n ) ∈ U, t ∈ T. Let f : T × U → R n ∈ C k rd , where C k rd denotes the set of all k-th rd-continuous functions, for each (t, x) ∈ T × U .
We define the averaged function F (·, ε) on time scales by Now, we are in the position to state our main result on the existence of periodic solutions to arbitrary order in ε for nonlinear dynamic equations on time scales.
is an open bounded set. We consider the following dynamic equation where f i : T × U → R n for i = 1, · · · , k, r : T × U × (−ε 0 , ε 0 ) are rd−continuous functions, T periodic in the first argument and locally Lipschitz with respect to x. Moreover, we assume the following conditions hold: (i).For each t ∈ T, p ∈ ∂U , there exists a neighborhood N p of p, a constant σ > 0 independent of ε and integers 1 ≤ j ≤ n such that where F (·, ε) is defined as (2).
Then, there exists a T −periodic solution x(t) of equation (3) such that x(t) ∈ U , for |ε| > 0 sufficiently small.
The rest of this paper is organized as follows. In Section 2, we give the basic definitions and properties related to the time scales briefly. In Section 3, we present proof of our main result via topological degree. In last Section, we present the first order averaging theorem for computing periodic solutions of equation (1). Finally, to illustrate our main result, the last section is devoted to some examples.

2.
Preliminaries. In this section, we will present some basic definitions, concepts and results concerning time scales which will be essential to prove our main results. For more details about time scales, see [4,5]. Let T be a time scale, that is, a closed and nonempty subset of R.
Definition 2.1. For t ∈ T, we define the forward jump operator and backward jump operator σ, ρ : T → T, respectively, as follows: In this definition, we put inf ∅ = sup T and sup ∅ = inf T. If σ(t) > t, we say that t is right-scattered. If σ(t) = t, then t is called right-dense. Analogously, if ρ(t) < t, we say that t is left-scattered. If ρ(t) = t, then t is called left-dense.
And we define the set T κ which is derived from T as follows: if T has a left-scattered maximum m, then T κ = T − {m}. Otherwise, T κ = T.
3. Assume f : T → R n is a function and let t ∈ T κ . Then we define f ∆ (t) to be the vector (provided it exists) with the property that for any ε > 0, there is a neighborhood U of t such that We call f ∆ (t) the delta (or Hilger) derivative of f at t. Similarily, we can define the nabla-derivative of the function f : T → R n , for details, see [4] and [5].
Definition 2.4. A function f : T → R n is called rd-continuous provided it is continuous at right-dense points in T and its left-sided limits exist (finite) at all left-dense points in T. The set of rd-continuous functions f : T → R n will be denoted by Remark 1. F is continuous on T when f ∈ C rd (T) (see Theorem 1.16 in [4]).
Next we will introduce the definition of periodic time scale.
For example, the time scale 3. Proof of Theorem 1.1. The following definition of a retraction map is a useful tool in the proof of our main result.
Definition 3.1. Let X be a topological space and A a subspace of X. Then a continuous map α : Proof. For simplicity, we denote Now we consider the equation where F : T×U ×(−ε 0 , ε 0 ) → R n is rd-continuous, T -periodic in t, locally Lipschitz in x, and T is a T -periodic time scale, U is an open subset of R n . Let x(t) be the solution of (3). By the existence and uniqueness theorem on time scales (see, Section 8.3 of [4]), there exists an ε 0 > 0 such that for any y 0 ∈Ū , and ε ∈ [−ε 0 , ε 0 ], the solution x(t) exists for all t ∈ [0, T ] T , and we have By Lemma 2.1 in [15], we know that the existence of T -periodic solution of (3) is equivalent to the following boundary value problem of (6).
, and define the norm as y = sup t∈[0,T ] T |y(t)|. We can see that X is a Banach space with the norm · .
For y 0 ∈ R n and x ∈ X with x(t) ∈Ū for all t ∈ [0, T ] T , we define an operator T (y 0 , x) as follows: Next, the frame of the proof is the same to Theorem 1.1 in [15]. The only difference is in the proof of 0 / . Now we give a detailed proof of it and we will omit other details.
By (a) and (b), we obtain Therefore, by the homotopy invariance, the theory of topological degree and Lemma 5.3, we have Then, it means that there is ( x * , y * ) ∈ U × V , such that A similar proof in (b) yields that x * (t) ∈ X 1 , That is Hence, from (16) and (17), we obtain that ( x * , y * ) is a fixed point of T in X. Thus x * (t) is a solution of boundary value problem (6). This completes the proof of Theorem 1.1.

Remark 2.
It is certain that the Theorem 1.1 remains effective for nabla dynamic equations on time scales. Namely, we can prove analogously that the nabla dynamic equation where f i : T × U → R n for i = 1, · · · , k, r : T × U × (−ε 0 , ε 0 ) are rd−continuous functions, T periodic in the first argument and locally Lipschitz with respect to x, under similar conditions to the ones presented on Theorem 1.1.
Remark 3. Slaík in [14] studied the averaging theory of first order on the existence of periodic solutions for dynamic equations on time scales by implicit function theorem. Recently, Llibre, Novaes, and Teixeira in [12] extended the averaging theory up to any order in ε by using the Brouwer degree. Comparing with their results, we add the condition (i) in Theorem 1.1. Our main results extend the averaging theory for dynamic equations on time scales to arbitrary order in ε. Moreover, our averaging functions F (·, ε) are much easier to calculate than them.

Applications.
Here we state the first order averaging method. Our proof is different from the one given in [14] (see Theorem 4.4).
Then there exists a T -periodic solution of equation (18) such that x(t) ∈ U , for |ε| > 0 sufficiently small.
Proof. The idea of the proof is the same for the previous Theorem 1.1. The only difference is in the proof of 0 / ∈ (id − H)(∂(U × V ) × [0, 1]). Now we give a detailed proof of it and we will omit other details.
(b)When λ ∈ (0, 1], as (id − H)( y 0 , x(t), λ) = 0, we have y 0 Note that By the definiton of X λ , we obtain x ∈ X λ , which means that α λ • x = x. Therefore, and Then where M 0 is a positive constant. Since f 1 (t, x) is locally Lipschitz with respect to x, we have where L is the local Lipchitz constant. Then If y 0 ∈ ∂U , from (20) we have This is contradictory to (26), when we let ε → 0. Thus y 0 / ∈ ∂U . We claim that x(t) ∈ ∂V , that we have x(t) ∈ ∂U for some t ∈ [0, T ] T . This is contradictory to (24), when we let ε → 0.
By (a) and (b), we obtain The conclution follows by applying Theorem 1.1.
Next we present some examples and applications of our main results.

4.2.
Example. Consider the dynamic equation where b, c, r ∈ C rd (T, R) and b, c, r are T -periodic with t. Since | tanh( Then all hypotheses of Theorem 4.1 are satisfied, and hence equation (28) has a T −periodic solution.
The following lemma (see, [4]) is useful in the calculation of examples.
The following lemma due to [7] plays a crucial role in the proof of main results.