EXISTENCE AND EXPONENTIAL STABILITY OF POSITIVE PERIODIC SOLUTIONS FOR SECOND-ORDER DYNAMIC EQUATIONS

In this article, we establish the existence of positive periodic solutions for second-order dynamic equations on time scales. The main method used here is the Schauder fixed point theorem. The exponential stability of positive periodic solutions is also studied. The results obtained here extend some results in the literature. An example is also given to illustrate this work.


Introduction
Time scales theory was initiated by Stefan Hilger in 1988 as a means of unifying theories from discrete analysis and continuous analysis. Difference equations are defined on discrete sets while differential equations are defined on an interval of the set of real numbers. However, dynamic equations on time scales are very important in the physical applications because they are either difference equations, differential equations or a combination of both. This means that dynamic equations are defined on discrete, connected or combination of both types of sets. Hence, the theory of time scales provides an extension of difference analysis and differential analysis, see [6,7,15,17] and the references therein.
Let T be a periodic time scale such that t 0 ∈ T. In this paper, we are interested in the positivity, periodicity and exponential stability of solutions of second-order dynamic equations. Inspired and motivated by the references in this paper, we consider the following second-order dynamic equation (1.1) with x ∆ (t 0 ) + ax σ (t 0 ) = 0 and x(t 0 ) = 1. Throughout this paper we assume that a ≥ 0, q, r ∈ C rd ([t 0 , ∞) ∩ T, R), α, β ∈ (0, ∞). To prove the positivity and periodicity of solutions of (1.1), we convert (1.1) into an equivalent integral equation and then employ the Schauder fixed point theorem. The sufficient conditions for the exponential stability of positive solutions are also considered.
In the special case T = R, Dorociakova, Michalkova, Olach and Saga in [13] show the existence and the exponential stability of positive solutions of (1.1). Then, the results presented in this paper extend the main results in [13]. The rest of this work is organized as follows. In Section 2, we present some basic concepts concerning the calculus on time scales that will be used to show our main results. We give some properties of the exponential function on a time scale as well as the Schauder fixed point theorem. We refer the reader to the monograph [18] for more details on the Schauder theorem. In Section 3, we prove our main results for the existence of positive periodic solutions by using the Schauder theorem, and we give an example to illustrate our existence results. In Section 4, we study the exponential stability of a positive periodic solution of (1.1). In Section 5, we establish new sufficient conditions for the existence and the exponential stability for a pipe-tank flow configuration.
Definition 1 [6]. A time scale T is an arbitrary nonempty closed subset of R.
The definition of periodic time scales was introduced by Kaufmann and Raffoul [16]. The following two definitions are found in [16].

Definition 2.
A time scale T is said to be periodic provided there exists a T > 0 such that if t ∈ T then t ± T ∈ T. For T = R, the period of the time scale is the smallest positive T . Example 1 [16]. The following time scales are periodic.
where 0 < q < 1 and N 0 is the natural numbers with zero, has period T = 1.
Remark 1 [16]. All periodic time scales are unbounded above and below.
Definition 3. Let T = R be a periodic time scale with period T . The function f : T → R is said to be periodic with period ω provided there exists a natural number n such that ω = nT , f (t ± ω) = f (t) for all t ∈ T and ω is the smallest number such that f (t ± ω) = f (t). If T = R, f is said to be periodic with period ω > 0 provided ω is the smallest positive number such that f (t ± ω) = f (t) for all t ∈ T. Definition 4 [6]. Let T be a time scale. The forward jump operator σ : T → T is defined by σ (t) = inf {s ∈ T : s > t} for all t ∈ T, while the graininess function µ : T → [0, ∞) is defined by Remark 2 [16]. Let T be a periodic time scale with period T . Then, the forward jump operator σ satisfies σ(t ± nT ) = σ(t) ± nT . Hence, µ(t ± nT ) = σ(t ± nT ) − (t ± nT ) = σ(t) − t = µ(t). So, µ is a periodic function with period T .
Definition 5 [6]. We say that the function f : T → R is regulated if its right-sided limits exist at all right-dense points in T and its left-sided limits exist at all left-dense points in T.
Definition 6 [6]. We say that the function f : T → R is rd-continuous if it is continuous at every right-dense point t ∈ T and its left-sided limits exist, and is finite at every left-dense point t ∈ T. We denote the set of rd-continuous functions f : T → R by We denote the set of differentiable functions f : T → R and whose derivative is rd-continuous by We say that the function f ∆ : T k → R is the delta derivative of f on T k .
Definition 8 [6]. A function p : T → R is said to be regressive if 1 + µ(t)p(t) = 0 for all t ∈ T. We denote the set of all rd-continuous and regressive functions p : T → R by R = R(T, R). We define the set R + of all rd-continuous and positively regressive functions by Theorem 1 [6]. Suppose f : T → R is a regulated function. Then there exists a function F which is pre-differentiable with region of differentiation D such that Definition 9 [6]. Suppose f : T → R is a regulated function. We say that the function F as in Theorem 1 is a pre-antiderivative of f . The indefinite integral of a regulated function f is defined by where F is a pre-antiderivative of f and C is an arbitrary constant. The Cauchy integral is defined by We say that a function F : T → R is an antiderivative of f : Theorem 2 [6]. Every rd-continuous function has an antiderivative.
Definition 10 [6]. For p ∈ R, we define the generalized exponential function e p as the unique solution of the initial value problem We give an explicit formula for e p (t, s) by with log is the principal logarithm function.
Lemma 1 [6]. For p, q ∈ R, we define the functions p ⊕ q and ⊖p by and which are elements of R.
Lemma 2 [6]. Let p, q ∈ R. Then (i) e 0 (t, s) ≡ 1 and e p (t, t) ≡ 1, (v) e p (t, s)e p (s, r) = e p (t, r), The following Schauder fixed point theorem plays important role to prove the existence results in the next section.

Positive periodic solutions
Next theorem guarantee the existence of positive ω-periodic solutions of (1.1).
Theorem 4. Assume that there exist positive constants m and M , and a rd-continuous func- and and define the operator S : Ω → X as follows for t ≥ t 0 . We will prove that SΩ ⊂ Ω. By using (3.1), for every x ∈ Ω and t ≥ t 0 we obtain Also for x ∈ Ω and t ≥ t 0 we have From (3.3), for every x ∈ Ω and t ≥ t 0 we obtain Finally we will prove that for x ∈ Ω, t ≥ t 0 the function Sx is ω-periodic. By using (3.2), for x ∈ Ω and t ≥ t 0 we get Now, we need to prove that the mapping S is completely continuous. So we will show that the mapping S is continuous. Let x i ∈ Ω be such that x i −→ x ∈ Ω as i −→ ∞. For t ≥ t 0 , we have By applying the Lebesgue dominated convergence theorem we obtain that Therefore S is continuous.
Next, we are going to prove that SΩ is relatively compact by applying the Arzela-Ascoli theorem. The uniform boundedness of SΩ follows from the definition of Ω. For t ≥ t 0 and x ∈ Ω we have which implies that the family SΩ is equicontinuous. By using the Arzela-Ascoli theorem SΩ is relatively compact. Therefore, S is completely continuous. By Theorem 3 there is an x 0 ∈ Ω such that Sx 0 = x 0 . We see that x 0 is a positive ω-periodic solution of (1.1). The proof is complete.
To illustrate the applications of Theorem 4 we give the following example.
Example 2. Consider the dynamic equation on T = πZ then µ(t) = π, Then for the conditions (3.1), (3.2) and ω = 4π we obtain We take m = e −1 and M = e, then Also, we put All conditions of Theorem 4 are satisfied. Thus (3.4) has a positive ω = 4π-periodic solution with x(t 0 ) = 1 and

Exponential stability of positive periodic solutions
In this section, we will prove the exponential stability of a positive ω-periodic solution of (1.1). Let x 1 be the positive ω-periodic solution of (1.1) with the initial condition x 1 (t 0 ) = 1 and x ∆ 1 (t 0 ) + ax σ 1 (t 0 ) = 0. Let x be the another positive ω-periodic solution of (1.1) with the initial condition x (t 0 ) = c 1 > 0, c 1 = 1 and x ∆ (t 0 ) + ax σ (t 0 ) = 0. Let After integration of (1.1), we obtain In a similar way one can easily show that Therefore This implies By using the mean value theorem, we get For m ≤ x (t) ≤ M , we suppose that the function is Lipschitzian in second argument.
Definition 11. Assume that x 1 is the positive ω-periodic solution of (1.1). If there exist positive constants K x 1 and λ for every positive ω-periodic solution x of (1.1) such that then x 1 is said to be exponentially stable.
In the next theorem, we prove the exponential stability of the positive periodic solution x 1 of (1.1).
We prove that there exists λ ∈ (0, ∞) such that where K x 1 = e λ (t 0 , 0) |y(t 0 )| + 1. We define the Lyapunov function For t > t 0 , we assume that L(t) < K x 1 . On the other hand there exists t * ≥ t 0 such that L(t * ) = K x 1 and L(t) < K x 1 for t ∈ [t 0 , t * ). By calculation of the upper left delta derivative of L(t) along the solution of (4.1), we get (L(t)) ∆ − ≤ −a |y σ (t)| e λ (t, 0) + e λ (t, 0) For t = t * we have If y(t) > 0, t ≥ t 0 , then from (4.1) it follows that, for t ≥ t 0 , the function y is decreasing. If y(t) < 0, t ≥ t 0 , then y is increasing for t ≥ t 0 . We conclude that |y(t)| , t ≥ t 0 has decreasing character. Then we obtain which is a contradiction. Hence, we get |y(t)| e λ (t, 0) < K x 1 for t ≥ t 0 and some λ ∈ (0, a) .
The proof is complete.

Application in a pipe-tank configuration
In [11], Cid et al. reformulated the problem of fluid motion in the pipe into the following periodic boundary value problem where a ≥ 0, b > 1, c > 0 and e is ω-periodic continuous on R. By using the change of variables u = x 1/(b+1) , the singular problem (5.1) can be transformed to the following regular problem where with 0 < α < β < 1. We will give new sufficient conditions ensuring the existence and the exponential stability of positive ω-periodic solutions of the following dynamic equation With respect to Theorems 4 and 5, we obtain the following theorem.
Then (5.2) has a positive ω-periodic solution which is exponentially stable.

Conclusion
In this paper, we provided the existence and exponential stability of positive periodic solutions with sufficient conditions for second-order dynamic equations on time scales. The main tools of this paper are the fixed point method and the Lyapunov method. However, by introducing new fixed mappings and suitable Lyapunov functionals, we get new existence and exponential stability conditions. An example illustrating our results is presented. The obtained results have a contribution to the related literature, and they improve and extend the results in [13] from the case of second-order differential equations to that case with second-order dynamic equations on time scales. It seems that the results of this paper can be extended to cover the case of delay second-order dynamic equations.