Existence and global exponential stability of periodic solutions for n-dimensional neutral dynamic equations on time scales

In this paper, by using the existence of the exponential dichotomy of linear dynamic equations on time scales and the theory of calculus on time scales, we study the existence and global exponential stability of periodic solutions for a class of n-dimensional neutral dynamic equations on time scales. We also present an example to illustrate the feasibility of our results. The results of this paper are completely new and complementary to the previously known results even in both the case of differential equations (time scale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}={\mathbb {R}}$$\end{document}T=R) and the case of difference equations (time scale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}={\mathbb {Z}}$$\end{document}T=Z).

reactions, many authors have studied the existence of solutions of various neutral delay models (Abbas and Bahuguna 2008;Ardjouni and Djoudi 2012;Chen and Lin 2010;Hovhannisyan 2014;Kaufmann and Raffoul 2006;Li and Saker 2014;Xu et al. 2007;Zhang et al. 2009). However, to the best of our knowledge, there are few papers published on the existence and stability of periodic solutions to neutral dynamic equations on time scales. Motivated by the above discussion, in this paper, we are concerned with the following neutral dynamic equation on time scales: where T is an ω-periodic time scale and satisfies that for t, s ∈ T, t + s ∈ T , A(t) = (a ij (t)) n×n is a regressive and rd-continuous matrix-valued function, f ∈ C rd (T × BC × BC, R n ) and f (t, x t , x � t ) is ω-periodic whenever x is a -differentiable ω-periodic function with rd-continuous -derivative, where BC denotes the Banach space of all bounded rd-continuous functions ϕ : [−θ , 0] ∩ T → R n with the norm |ϕ| 0 = max 1≤i≤n sup s∈[−θ ,0]∩T |ϕ i (s)| where ϕ = (ϕ 1 , φ 2 , . . . , ϕ n ) T , ω > 0 is a constant, θ is a positive number or ∞ and if θ = ∞, then we set [−θ, 0] = (−∞, 0]. If x, x � ∈ C rd (T, R n ), then for any t ∈ T, x t and x t ∈ BC are defined by x t (s) = x(t + s) and x � t (s) = x � (t + s) for s ∈ [−θ , 0] ∩ T, respectively.
Remark 1 Throughout this paper, we denote the class of all functions f : T × BC × BC → R n that are rd-continuous with respect to their first argument and continuous with respect to their second and third arguments by C rd (T × BC × BC, R n ).
Remark 2 If θ is a finite positive number, then Eq. (1) is a bounded delay neutral dynamic equation on time scales and if θ is infinite then Eq. (1) is a unbounded delay neutral dynamic equation on time scales.
Our main purpose of this paper is to study the existence and global exponential stability of periodic solutions for (1) by using the exponential dichotomy of linear dynamic equations and the theory of calculus on time scales. As we all know, Eq. (1) contains many differential equation models and difference equation models as its special cases. For example, if we take where ϕ = (ϕ 1 , ϕ 2 , . . . , ϕ n ), then (1) reduces to the following neural network with neutral type delays: which was studied in Li et al. (2012). If we take then (1) reduces to which was studied in Liu and Li (2004). Even in both the case of differential equations (time scale T = R) and the case of difference equations (T = Z), our results are completely new and complementary to the previously known results.
For an rd-continuous ω-periodic function u : T → R n , we define |u| 0 = max 1≤i≤n max t∈[0,ω] T |u i (t)|. For matrices or vectors A, B, A ≥ B (or A > B) means that all entries of A are greater than or equal to (or greater than) corresponding entries of B. For A(t) = (a ij (t)) n×n , we can take ||A|| = max 1≤i≤n n j=1 |a + ij |. The initial condition of (1) is . Throughout this paper, we assume that the following condition holds: is ω-periodic with respect to its first argument and there exist positive constants L 1 , L 2 such that for all t ∈ T and ϕ i , ψ i ∈ BC, i = 1, 2.

Preliminaries
In this section, we introduce some definitions and state some preliminary results. Let T be a nonempty closed subset (time scale) of R. The forward and backward jump operators σ , ρ : T → T and the graininess µ : T → R + are defined, respectively, by right-dense if t < sup T and σ (t) = t, and right-scattered if σ (t) > t. If T has a left-scattered maximum m, then T k = T\{m}; otherwise T k = T. If T has a right-scattered minimum m, then T k = T\{m}; otherwise T k = T.
A function f : T → R is right-dense continuous provided it is continuous at rightdense points in T and its left-side limits exist at left-dense points in T. If f is continuous at each right-dense point and each left-dense point, then f is said to be continuous on T.
We denote the class of all rd-continuous functions f : T → R by C rd (T, R).
For y : T → R and t ∈ T k , we define the delta derivative of y(t), y � (t), to be the number (if it exists) with the property that for a given ε > 0, there exists a neighborhood U of t such that for all s ∈ U.
We denote the class of all -differentiable functions with rd-continuous -derivative f : T → R by C 1 rd (T, R). If y is continuous, then y is right-dense continuous, and if y is -differentiable at t, then y is continuous at t.
Let y be right-dense continuous. If Y � (t) = y(t), then we define the delta integral by

Definition 1 (Bohner and Peterson 2001) Let
Definition 2 (Kaufmann and Raffoul 2006) We say that a time scale T is periodic if there exists p > 0 such that if t ∈ T, then t ± p ∈ T. For T � = R, the smallest positive p is called the period of the time scale.
Definition 3 (Kaufmann and Raffoul 2006) Let T � = R be a periodic time scale with period p. We say that the function f : T → R is periodic with period ω if there exists a natural number n such that ω = np, f (t + ω) = f (t) for all t ∈ T and ω is the smallest positive number such that f (t + ω) = f (t).

Definition 4 (Bohner and Peterson 2001)
Definition 5 (Bohner and Peterson 2001) Let A, B be two n × n-matrix-valued regressive functions on T, we define for all t ∈ T k .
Definition 6 (Bohner and Peterson 2001) Let t 0 ∈ T and assume that A ∈ R is a n × n -matrix-valued function. The unique matrix-valued solution of the initial value problem where, I denotes as usual the n × n-identity matrix, is called the matrix exponential function (at t 0 ) and it is denoted by e A (·, t 0 ).
Lemma 1 (Bohner and Peterson 2001) Let A ∈ R be a n × n-matrix-valued functions on T and suppose that f : T → R n is rd-continuous. Let t 0 ∈ T and x 0 ∈ R n . Then the initial value problem has a unique solution x : T → R n , which is given by  ) and B(t) commute. Lemma 3 (Bohner and Peterson 2001) If A ∈ R and a, b, c ∈ T, then and Definition 7 (Zhang et al. 2010a) Let x ∈ R n and A(t) be a n × n matrix-valued function on T, the linear system is said to admit an exponential dichotomy on T if there exist positive constants k i , α i , i = 1, 2, projection P and the fundamental solution matrix X(t) of (2) satisfying Lemma 4 (Zhang et al. 2010a) If (2) admits an exponential dichotomy, then the following ω-periodic system: has an ω-periodic solution as follows: where X(t) is the fundamental solution matrix of (2).

Lemma 5 (Zhang et al. 2010a) If A(t) is a uniformly bounded rd-continuous n × n matrix-valued function on
T and there is a δ > 0 such that then (2) admits an exponential dichotomy on T.
Definition 8 Let x(t) be an ω-periodic solution of (1) with initial value ϕ(s). If there exists a positive constant with − ∈ R + such that for t 0 ∈ [−θ , 0] T , there exists M > 1 such that for an arbitrary solution y(t) of (1) with initial value ψ(s) satisfies Then the solution x(t) is said to be globally exponentially stable.
Proof By (H 3 ), we can take a positive constant L satisfying where a = |f (·, 0, 0)| 0 . We set X 0 = {ϕ ∈ X| ||ϕ|| X ≤ L}. For any given ϕ ∈ X 0 , we consider the following periodic system: Since (H 2 ) holds, by Lemma 4, we obtain that (3) has an ω-periodic solution, which is expressed as follows: For ϕ ∈ X 0 , define the following operator: First we show that for any ϕ ∈ X 0 , we have �ϕ ∈ X 0 . Note that

So, we have that
On the other hand, we have (1 + µ(s) ⊖ α 2 )e ⊖α 2 (s, t)�s ⊖α 2 e ⊖α 2 (s, t)�s Hence, we have ||�ϕ|| X ≤ L, that is, �ϕ ∈ X 0 . Next, we show that is a contraction. For any ϕ, ψ ∈ X 0 , we have and By (H 3 ), we have ||�ϕ − �ψ|| X ≤ q�ϕ − ψ� X . It follows that is a contraction. Therefore, according to the Banach fixed-point theorem, has a fixed point in X 0 , that is, (1) has a unique periodic solution in X 0 . This completes the proof of Theorem 1. In view of Lemma 5 and Theorem 1, we have the following corollary: there is a constant δ > 0 such that Then (1) has a unique ω-periodic solution.

Global exponential stability of periodic solution
(H 4 ) L 1 + L 2 + ||A|| N < 1 and N (L 1 + L 2 )(1 + ϑ)||A|| < α. Then the periodic solution of (1) is globally exponentially stable. Proof By Theorem 1, (1) has an ω-periodic solution x(t) with the initial value ϕ(s) . Suppose that y(t) is an arbitrary solution of (1) with the initial value ψ(s). Denote z(t) = y(t) − x(t). Then it follows from (1) that for t ∈ T, The initial value condition of (4) is By Lemma 1, for t 0 ∈ [−θ , 0) T with t 0 < t, we have Take a constant 0 < < α with − ∈ R + and let By (H 4 ), it is easy to verify that M > 1 and hence, we have We claim that To prove this claim, we show that for any constant p > 1, the following inequality holds which means that and By way of contradiction, assume that (7) does not hold. We will have the following three cases. Case One: (9) is true and (8) is not true. Then there exists t 1 ∈ (t 0 , +∞) T such that Hence, there must be a constant c ≥ 1 such that Then, by (5), for t = t 1 , we have (6) ||z|| X ≤ M||φ|| X e ⊖ (t, t 0 ), ∀t ∈ (t 0 , +∞) T .
Case Three: (8) and (9) are both untrue. By Case One and Case Two, we can obtain a contradiction. Therefore, (7) holds. Let p → 1, (6) holds. Hence, we have that which implies that the periodic solution x(t) of (1) is globally exponentially stable. This completes the proof of Theorem 2.
Remark 4 Example (10) shows that both the continuous case of (10) and its discrete analogue have the same dynamical property for the periodic case.