Periodic and asymptotically periodic solutions in nonlinear coupled Volterra integro-dynamic systems with in nite delay on time scales

Let T be a periodic time scale. The purpose of this paper is to use Schauder's xed point theorem to prove the existence of periodic and asymptotically periodic solutions of nonlinear coupled Volterra integro-dynamic systems with in nite delay on time scales. The results obtained here extend the work of Ra oul [22].


Introduction
Time scales calculus was initiated in 1988 by Stefan Hilger. It bridges the gap between continuous and discrete analysis and expands on both theories. Dierential equations are dened on an interval of the set of real numbers while dierence equations are dened on discrete sets. However, some physical systems are modeled by what is called dynamic equations because they are either dierential equations, dierence equations or a combination of both. This means that dynamic equations are dened on connected, discrete or combination of both types of sets. Hence, time scales calculus provides a generalization of dierential and dierence analysis, see [9,10,18,20] and the references therein.
Delay dynamic equations arise from a variety of applications including in various elds of science and engineering such as applied sciences, practical problems concerning mechanics, the engineering technique elds, economy, control systems, physics, chemistry, biology, medicine, atomic energy, information theory, harmonic oscillator, nonlinear oscillations, conservative systems, stability and instability of geodesic on Riemannian manifolds, dynamics in Hamiltonian systems, etc, see [9,10,20,25,26]. In particular, problems concerning qualitative analysis of delay dynamic equations have received the attention of many authors, see [1] [22], [24] [26] and the references therein.
Let T be a periodic time scale such that 0 ∈ T. In this article, we are interested in the analysis of qualitative theory of periodic and asymptotically periodic solutions of coupled Volterra integro-dynamic equations. Inspired and motivated by the references in this paper, we consider the following nonlinear coupled Volterra integro-dynamic systems with innite delay where h 1 , h 2 , p 1 , p 2 , a and b are rd-continuous functions, f and g are continuous functions. To show the existence of periodic and asymptotically periodic solutions of (1), we transform (1) into an integral system and then use Schauder's xed point theorem. In the special case T = R, Raoul in [22] show the existence of periodic and asymptotically periodic solutions of (1). Then, the results presented in this paper extend the main results in [22].

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 (see [1]- [13], [17]- [20] and papers therein). The theory of dynamic equations unies the theories of dierential equations and dierence equations. We suppose that the reader is familiar with the basic concepts concerning the calculus on time scales for dynamic equations. Otherwise one can nd in Bohner and Peterson books [9,10,20] most of the material needed to read this paper. We start by giving some denitions necessary for our work. The notion of periodic time scales is introduced in Kaufmann and Raoul [19]. The following two denitions are borrowed from [19].
Remark 2.1 ([19]). All periodic time scales are unbounded above and below. Denition 2.2. Let T = R be a periodic time scale with period ω. We say that the function f : T → R is periodic with period T if there exists a natural number n such that T = nω, f (t ± T ) = f (t) for all t ∈ T and T is the smallest number such that f (t ± T ) = f (t).
If T = R, we say that f is periodic with period T > 0 if T is the smallest positive number such that f (t ± T ) = f (t) for all t ∈ T. Remark 2.2 ([19]). If T is a periodic time scale with period ω, then σ(t ± nω) = σ(t) ± nω. Consequently, the graininess function µ satises µ(t ± nω) = σ(t ± nω) − (t ± nω) = σ(t) − t = µ(t) and so, is a periodic function with period ω. Denition 2.3 ([9]). A function f : T → R is called rd-continuous provided it is continuous at every rightdense point t ∈ T and its left-sided limits exist, and is nite at every left-dense point t ∈ T. The set of rd-continuous functions f : T → R will be denoted by The set of functions f : T → R that are dierentiable and whose derivative is rd-continuous is denoted by Denition 2.4 ([9]). For f : T → R, we dene 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 The function f ∆ : T k → R is called the delta (or Hilger) derivative of f on T k . Denition 2.5 ([9]). A function p : T → R is called regressive provided 1 + µ(t)p(t) = 0 for all t ∈ T. The set of all regressive and rd-continuous functions p : T → R will be denoted by R = R(T, R). We dene the set R + of all positively regressive elements of R by Denition 2.6 ([9]). Let p ∈ R, then the generalized exponential function e p is dened as the unique solution of the initial value problem x ∆ (t) = p(t)x(t), x(s) = 1, where s ∈ T. An explicit formula for e p (t, s) is given by where log is the principal logarithm function.
Lemma 2.1 ([9]). Let p, q ∈ R. Then The proof of the main results in the next section is based upon an application of the following Schauder xed point theorem. Theorem 2.1 (Schauder's xed point theorem [23]). Let X be a Banach space, and Ω be a convex closed bounded subset of X. If E : Ω → Ω is completely continuous, then E has at least one xed point in Ω.
Denition 2.7. A map is completely continuous if it is continuous and it maps bounded sets into relatively compact sets.

Periodic solutions
Let T > 0, T ∈ T be xed and if T = R, T = nω for some n ∈ N. By the notation [a, b] we mean where both ϕ and ψ are real valued rd-continuous functions on T. Then P T is a Banach space when endowed with the maximum norm see [3]. Throughout this paper, we assume that h 1 , p 1 ∈ R + , h 1 , p 1 , h 2 , p 2 , a and b are rd-continuous functions, f and g are continuous functions, and for all t ∈ T. Also, we assume that e h 1 (T, 0) = 1, e p 1 (T, 0) = 1.
The following lemma is fundamental to our results. Lemma 3.1. Assume (2) and (3) hold. If (x, y) ∈ P T , then (x, y) is a solution of (1) if and only if and Proof. Let (x, y) ∈ P T be a solution of (1). First we write the rst equation of (1) as Multiplying both side by e h 1 (σ(t), 0) and then integrate from t to t + T to obtain Periodicity of x gives Thus In the similar fashion The proof is complete by reversing every step.
Since h 1 , h 2 , p 1 and p 2 are rd-continuous T -periodic functions, then there exist positive constants H 1 , H 2 , P 1 and P 2 such that |h i (t)| ≤ H i and |p i (t)| ≤ P i for i = 1, 2. Let L 1 and L 2 be positive constants such that 0 < L 1 H 2 T < 1 and 0 < L 2 P 2 T < 1. Also, assume there exist positive constants M 1 , M 2 , A and B such that |g(x, y)| ≤ M 2 , e h 1 (t + T, σ(u)) Set We dene a subset Ω x,y of P T as follows Then Ω x,y is a bounded closed convex subset of P T . Now for (x, y) ∈ Ω x,y we can dene an operator and Theorem 3.1. Suppose (2), (3) and (6) Proof. It is clear from Lemma 3.1 that E 1 (x, y)(t + T ) = E 1 (x, y)(t) and E 2 (x, y)(t + T ) = E 2 (x, y)(t).
Thus, E maps Ω x,y into itself, i.e E (Ω x,y ) ⊆ Ω x,y . Now, we shall prove that E is continuous. Let x l , y l be a sequence in Ω x,y such that lim l→∞ x l , y l − (x, y) = 0.
Since Ω x,y is closed, we have (x, y) ∈ Ω x,y . Then by denition of E we have in which The continuity of f along with the Lebesgue dominated convergence theorem implies that In similar way we have This show that E is a continuous map. To show that the map E is completely continuous, we will show that E (Ω x,y ) is relatively compact. We know that E (Ω x,y ) ⊆ Ω x,y , which means E (Ω x,y ) is uniformly bounded because Ω x,y is uniformly bounded. Moreover, a direct calculation shows that Then, there exists a positive constant L such that This means that E(x, y) (t) ≤ L. Therefore the set E (Ω x,y ) is equicontinuous, and hence by Arzela-Ascoli's theorem, it is relatively compact. By Schauder's xed point theorem, we conclude that there exists (x, y) ∈ Ω x,y such that (x, y) = E(x, y). Now, we relax condition (7). (6) and (9)- (11) hold. In addition, we assume the existence of continuous nondecreasing function G such that |g(x, y)| ≤ g(|x| , y) ≤ QG(|x|) for some positive constant Q, and for u > 0 we ask that Then (1) has a T -periodic solution.
Proof. Set For (x, y) ∈ Ω x,y , we have from the proof of Theorem 3.1 that Thus, The rest of the proof follows along the lines of the proof of Theorem 3.1.
Proof. Set For (x, y) ∈ Ω x,y , we have from the proof of Theorem 3.1 that Thus, The rest of the proof follows along the lines of the proof of Theorem 3.1.

Asymptotically periodic solutions
In this section, we obtain asymptotically periodic solutions of (1).
Denition 4.1. A function x is called asymptotically T -periodic if there exist two functions x 1 and x 2 such that x 1 is T -periodic, lim t→∞ x 2 (t) = 0 and x(t) = x 1 (t) + x 2 (t) for all t ∈ T.
In this section we do not assume the periodicity condition on the functions h 2 , p 2 , a and b. We only assume h 1 , p 1 ∈ R + , h 1 and p 1 are T -periodic, and e h 1 (T, 0) = e p 1 (T, 0) = 1.
Since h 1 and p 1 are T -periodic, there are constants S k , s k , M * k , m k , k = 1, 2, such that Also, we assume that there are positive constants A * , B * , M * 3 and M * 4 such that In addition, we suppose that and Finally, we Assume that Theorem 4.1. Suppose that (6), (7) and (15)- (21) hold. Then the system (1) has asymptotically T -periodic solution (x, y) satisfying Proof. Dene Then P * T is a Banach space when endowed with the maximum norm Dene a subset Ω x,y of P * T as follows. For a positive constant W * to be dened later in the proof, let Then Ω x,y is a bounded closed convex subset of P * T . Now, for (x, y) ∈ Ω x,y we can dene an operator F : Ω x,y → P * T by F (x, y)(t) = (F 1 (x, y)(t), F 2 (x, y)(t)) , where We will show that the mapping F has a xed point in Ω x,y . Set We note that W * is well dened due to (21). First, we demonstrate that F (Ω x,y ) ⊆ Ω x,y . If (x, y) ∈ Ω x,y , then In similar way we have By letting We will prove that x 1 and y 1 are T -periodic. From (15), one can see Similarly, y 1 is T -periodic. Hence F (Ω xy ) ⊆ Ω xy . The proof that F is completely continuous is similar to the corresponding work in Theorem 3.1, hence we omit it here. Therefore, by Schauder's xed point theorem, there exists a xed point (x, y) ∈ Ω x,y such that (x, y) = F (x, y) = (F 1 (x, y), F 2 (x, y)) .
Now we show that this xed point is a solution of (1). Let Then a delta dierentiation with respect to t gives a(t, s)f (x(s), y(s))∆s, and y (t) = p 1 (t)x(t) + p 2 (t)y(t) + Then (x, y) is a solution of (1). Therefore, (x, y) given by y(t) = y 1 (t) + y 2 (t), is asymptotically T -periodic solution of (1).
Acknowledgements. The authors are grateful to the referees for their valuable comments which have led to improvement of the presentation.