Existence of Positive Periodic Solutions for Nonlinear Neutral Dynamic Equations with Variable Coefficients on a Time Scale

Let T be a periodic time scale. The purpose of this paper is to use Krasnosel’skĭı’s fixed point theorem to prove the existence of positive periodic solutions for nonlinear neutral dynamic equations with variable coefficients on a time scale. We invert these equations to construct a sum of a contraction and a compact map which is suitable for applying the Krasnosel’skĭı’s theorem. The results obtained here extend the work of Candan [11].


Introduction
Let T be a periodic time scale such that 0 ∈ T. In this paper, we are interested in the analysis of qualitative theory of positive periodic solutions of dynamic equations.Motivated by the papers [1]- [7], [10]- [20] and the references therein, we consider the following two kinds of nonlinear neutral dynamic equations with variable coefficients where x △ is the △-derivative on T (see [8]).Throughout this paper we assume that τ = mω if T has period ω and τ is fixed if T = R.Our purpose here is to use the Krasnosel'skiȋ's fixed point theorem to show the existence of positive periodic solutions on time scales for equation (1.1).To reach our desired end we have to transform (1.1) into integral equation written as a sum of two mapping; one is a contraction and the other is compact.After that, we use Krasnosel'skiȋ's fixed point theorem, to show the existence of a positive periodic solution for equation A. Ardjouni, A. Djoudi (1.1).In the special case T = R, in [11] we show that (1.1) have a positive periodic solution by using Krasnosel'skiȋ's fixed point theorem.The organization of this paper is as follows.In Section 2, we present some preliminary material that we will need through the remainder of the paper.We will state some facts about the exponential function on a time scale as well as the Krasnosel'skiȋ's fixed point theorem.For details on Krasnosel'skiȋ's theorem we refer the reader to [21].Also, we present the inversion of (1.1), and we give the Green's functions of (1.1), which play an important role in this paper.In Section 3 and Section 4, we present our main results on existence of positive periodic solutions of (1.1).The results presented in this paper extend the main results in [11].

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], [3], [5]- [9], [16], [17] 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 [8] and [9] 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 Atici et al. [6] and Kaufmann and Raffoul [16].The following two definitions are borrowed from [6] and [16].
Definition 2.1.We say that a time scale T is periodic if there exists a ω > 0 such that if t ∈ T then t ± ω ∈ T. For T = R, the smallest positive ω is called the period of the time scale.
Below are examples of periodic time scales taken from [16].
Definition 2.4.Let T = R be a periodic time scales 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 2.5 ( [16]
).If T is a periodic time scale with period ω, then σ (t ± nω) = σ (t)±nω.Consequently, the graininess function µ satisfies µ (t and so, is a periodic function with period ω. Our first two theorems concern the composition of two functions.The first theorem is the chain rule on time scales ( [8], Theorem 1.93).
Theorem 2.7 (Substitution).Assume ν : T → R is strictly increasing and Let p ∈ R and µ (t) = 0 for all t ∈ T. The exponential function on T is defined by It is well known that if p ∈ R + , then e p (t, s) > 0 for all t ∈ T. Also, the exponential function y (t) = e p (t, s) is the solution to the initial value problem y △ = p (t) y, y (s) = 1.Other properties of the exponential function are given in the following lemma.
and e p (t + T, t) is independent of t ∈ T whenever p ∈ R.

Corollary 2.11 ( [1]
).If p ∈ R + and p (t) < 0 for all t ∈ T, then for all s ∈ T with s ≤ t we have We state Krasnosel'skiȋ's fixed point theorem which enables us to prove the existence of positive periodic solutions to (1.1).For its proof we refer the reader to [21].
Theorem 2.12 (Krasnosel'skiȋ).Let D be a closed convex nonempty subset of a Banach space (B, .).Suppose that A and B map D into B such that We will need the following lemma whose proof can be found in [16].
In this paper we assume that a ∈ R + , c are continuous and for all t ∈ T, where c △ is continuous.Also, we assume We also assume that the function f (t, x) is continuous in their respective arguments and periodic in t with period T , that is, The following lemma is essential for our results on existence of positive periodic solutions of (1.1).
Lemma 2.14.Suppose (2.2)-(2.4)hold.If x ∈ P T , then x is a solution of equation (1.1) if and only if Proof.Let x ∈ P T be a solution of (1.1).First we write this equation as Multiply both sides of the above equation by e a (t, 0) we get Since e a (t, 0) △ = a (t) e a (t, 0) we find Taking the integral from t to t + T , we obtain As a consequence, we arrive at Dividing both sides of the above equation by e a (t, 0) and using the fact that x (t + T ) = x (t), (2.2) and (2.4), we obtain Since each step is reversible, the converse follows easily.This completes the proof.✷ Corollary 2.15.Suppose c (t) = 0 for all t ∈ T and (2.2)-(2.4)hold.If x ∈ P T , then x is a solution of equation (1.1) if and only if where G is given by (2.6).
From Lemma 2.8 and Theorem 2.9, we have for all t, s ∈ R, (2.9)

Existence of positive periodic solutions in the case |c (t)| > 1
To apply Theorem 2.12, we need to define a Banach space B, a closed convex subset D of B and construct two mappings, one is a contraction and the other is compact.So, we let (B, • ) = (P T , • ) and D = {ϕ ∈ B : L ≤ ϕ ≤ K}, where L is a non-negative constant and K is a positive constant.We express equation (2.7) as where A 1 , B 1 : D → B are defined by In this section we obtain the existence of a positive periodic solution of (1.1) by considering the two cases; 1 < c (t) < ∞ and −∞ < c (t) < −1 for all t ∈ T. Denote In the case 1 < c (t) < ∞, we assume that there exist positive constants c 1 and c 2 such that ) and for all t ∈ [0, T ] , x, y ∈ D Proof.We first show that Use Theorem 2.7 with u = s − T to get From (2.2), (2.3) and (2.8), we obtain We show that A 1 (D) is uniformly bounded.For t ∈ [0, T ] and for ϕ ∈ D, we have by (2.9) and (3.5).Thus from the estimation of |(A 1 ϕ) (t)| we arrive This shows that A 1 (D) is uniformly bounded.It remains to show that A 1 (D) is equicontinuous.Let ϕ n ∈ D, where n is a positive integer.Next we calculate (A 1 ϕ n ) △ (t) and show that it is uniformly bounded.By making use of (2.2) and (2.4) we obtain by taking the derivative in (3.1) that Consequently, by invoking (2.9), (3.5) and Lemma 2.13, we obtain for some positive constant D. Hence the sequence (A 1 ϕ n ) is equicontinuous.The Ascoli-Arzela theorem implies that a subsequence Proof.Let B 1 be defined by (3.2).Obviously, B 1 ϕ is continuous and it is easy to show that (B 1 ϕ) (t + T ) = (B 1 ϕ) (t).So, for any ϕ, ψ ∈ D, we have Proof.By Lemma 3.1, the operator A 1 : D → B is compact and continuous.Also, from Lemma 3.2, the operator On the other hand, Clearly, all the hypotheses of the Krasnosel'skiȋ's theorem are satisfied.Thus there exists a fixed point x ∈ D such that x = A 1 x + B 1 x.By Lemma 2.14 this fixed point is a solution of (1.1).Next, we prove that x ∈ D 1 .We just need to prove that for all t ∈ [0, T ], x (t) > L. Otherwise, there exists t * ∈ [0, T ] satisfying x (t * ) = L. From (2.7), we have Noting that F (s, x, y) ≥ (c 2 − 1) L and F (t 0 , x) > (c 2 − 1) L, t 0 ∈ [0, T ], we obtain This is a contradiction.So, x ∈ D 1 .The proof is complete.✷ Remark 3.4.When T = R, Theorem 3.3 reduces to Theorem 1 of [11].
In the case −∞ < c (t) < −1, we substitute conditions (3.3)-(3.5)with the following conditions respectively.We assume that there exist a negative constants c 3 and c 4 such that and for all t ∈ [0, T ] , x, y ∈ D We express equation (2.5) as where A 2 , B 2 : D → B are defined by In this section we obtain the existence of a positive periodic solution of (1.1) by considering the two cases; 0 ≤ c (t) < 1 and −1 < c (t) ≤ 0 for all t ∈ T. Denote In the case 0 ≤ c (t) < 1, we assume that there exists positive constant c 1 such that and for all t ∈ In the case −1 < c (t) ≤ 0, we assume that there exists negative constant c 2 such that c 2 ≤ c (t) ≤ 0, for all t ∈ [0, T ] , (4.6) and for all t ∈ [0, T ] , x, y ∈ D A is compact and continuous, (iii) B is a contraction mapping.Then there exists z ∈ D with z = Az + Bz.Let T > 0, T ∈ T be fixed and if T = R, T = np for some n ∈ N. By the notation [a, b] we mean [a, b] = {t ∈ T : a ≤ t ≤ b} , unless otherwise specified.The intervals [a, b) , (a, b] and (a, b) are defined similarly.Define P T = {ϕ : T → R | ϕ ∈ C and ϕ (t + T ) = ϕ (t)} where C is the space of continuous real-valued functions on T. Then (P T , • ) is a Banach space with the supremum norm ϕ = sup t∈T |ϕ (t)| = sup t∈[0,T ] |ϕ (t)| .

Lemma 3 . 2 .
uniformly to a continuous T -periodic function.Thus A 1 is continuous and A 1 (D) is contained in a compact subset of B. ✷ Suppose that (2.2)-(2.4),(3.3) and (3.4) hold.Then B 1 : D → B is a contraction.