Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 11, 1-30; http://www.math.u-szeged.hu/ejqtde/

Linear impulsive dynamic systems on time scales

Vasile Lupulescu^{∗,}^{1}, Akbar Zada^{∗∗}

∗“Constantin Brancusi” University, Republicii 1, 210152 Targu-Jiu, Romania. E-mail: lupulescu v@yahoo.com

∗∗Government College University, Abdus Salam School of Mathematical Sciences, (ASSMS), Lahore, Pakistan. E-mail : zadababo@yahoo.com

Abstract. The purpose of this paper is to present the fundamental concepts of the basic theory for linear impulsive systems on time scales. First, we introduce the transition matrix for linear impulsive dynamic systems on time scales and we establish some properties of them. Second, we prove the existence and uniqueness of solutions for linear impulsive dynamic systems on time scales. Also we give some sufficient conditions for the stability of linear impulsive dynamic systems on time scales.

1 Introduction

Differential equations with impulse provide an adequate mathematical description of various real-word phenomena in physics, engineering, biology, economics, neutral network, social sciences, etc. Also, the theory of impulsive differential equations is much richer than the corresponding theory of differential equations without impulse effects. In the last fifty years the theory of impulsive differential equations has been studied by many authors. We refer to the monographs [10]-[12], [27], [33], [42] and the references therein.

S. Hilger [28] introduced the theory of time scales (measure chains) in order to create a theory that can unify continuous and discrete analysis. The theory of dynamic systems on time scales allows us to study both continuous and discrete dynamic systems simultaneously (see [8], [9], [28], [29]). Since Hilger’s initial work [28] there has been significant growth in the theory of dynamic systems on time scales, covering a variety of different qualitative aspects. We refer to the books [9], [15], [16], [32] and the papers [1], [3], [17], [20], [21], [24], [30], [34], [38], [40], [41]. In recent years, some authors studied impulsive dynamic systems on time scales [13], [25], [34], [36], but only few authors have studied linear impulsive dynamic systems on time scales.

In this paper we study some aspects of the qualitative theory of linear impulsive dynamic systems on time scales. In Section 2 we present some preliminary results on linear dynamic systems on time scales and also we give an impulsive inequality on time scales. In Section 3 we prove the existence and uniqueness of solutions for homogeneous linear impulsive dynamic systems on time scales. For this, we introduce the transition matrix for linear impulsive dynamic systems on time scales and we give some properties of the transition matrix. In Section 4 we prove the existence and uniqueness of solutions for nonhomogeneous linear impulsive dynamic systems on time scales. In Section 5 we give some sufficient conditions for the stability of linear impulsive dynamic systems on time scales. Finally, in Section 6 we present a brief summary of time scales analysis.

1Corresponding author

2 Preliminaries

Let ℝ^{n} be the space of n-dimensional column vectors x = col(x_{1}, x_{2}, ...x_{n}) with a norm ||·||. Also, by the same symbol ||·|| we will denote the corresponding matrix norm in the space M_{n}(ℝ) of n × n matrices. If A ∈ M_{n}(ℝ), then we denote by A^{T} its conjugate transpose. We recall that ||A|| := sup{||Ax||; ||x|| ≤ 1} and the following inequality ||Ax|| ≤ ||A|| · ||x|| holds for all A ∈ M_{n}(ℝ) and x ∈ ℝ^{n}. A time scales 𝕋 is a nonempty closed subset of ℝ. The set of all rd-continuous functions f : 𝕋 → ℝ^{n} will be denoted by C_{rd}(𝕋, ℝ^{n}).

The notations [a, b], [a, b), and so on, will denote time scales intervals such as [a, b] := {t ∈ 𝕋; a ≤ t ≤ b}, where a, b ∈ 𝕋. Also, for any τ ∈ 𝕋, let 𝕋_{(τ)} := [τ, ∞) ∩ 𝕋 and 𝕋_{+} := 𝕋_{(0)}. Then

BC_{rd}(𝕋_{(τ)}, ℝ^{n}) := {f ∈ C_{rd}(𝕋_{(τ)}, ℝ^{n}); sup_{t∈𝕋_{(τ)}}||f(t)|| < +∞}

is a Banach space with the norm ||f|| := sup_{t∈𝕋_{(τ)}}||f(t)||.

We denote by R (respectively R^{+}) the set of all regressive (respectively positively regressive) functions from 𝕋 to ℝ.

The space of all rd-continuous and regressive functions from 𝕋 to ℝ is denoted by C_{rd}R(𝕋, ℝ). Also,

C_{rd}^{+}R(𝕋, ℝ) := {p ∈ C_{rd}R(𝕋, ℝ); 1 + μ(t)p(t) > 0 for all t ∈ 𝕋}.

We denote by C_{rd}^{1}(𝕋, ℝ^{n}) the set of all functions f : 𝕋 → ℝ^{n} that are differentiable on 𝕋 and its delta-derivative f^{Δ}(t) ∈ C_{rd}(𝕋, ℝ^{n}). The set of rdcontinuous (respectively rd-continuous and regressive) functions A : 𝕋 → M_{n}(ℝ) is denoted by C_{rd}(𝕋, M_{n}(ℝ)) (respectively by C_{rd}R(𝕋, M_{n}(ℝ))). We recall that a matrix-valued function A is said to be regressive if I + μ(t)A(t) is invertible for all t ∈ 𝕋, where I is the n × n identity matrix.

Now consider the following dynamic system on time scales

x^{Δ} = A(t)x  (2.1)

where A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)). This is a homogeneous linear dynamic system on time scales that is nonautonomous, or time-variant. The corresponding nonhomogeneous linear dynamic system is given by

x^{Δ} = A(t)x + h(t)  (2.2)

where h ∈ C_{rd}(𝕋_{+}, ℝ^{n}).

A function x ∈ C_{rd}^{1}(𝕋_{+}, ℝ^{n}) is said to be a solution of (2.2) on 𝕋_{+} provided x^{Δ}(t) = A(t)x(t) + h(t) for all t ∈ 𝕋_{+}.

Theorem 2.1. (Existence and Uniqueness Theorem [15, Theorem 5.8]) If A ∈ C_{rd}ℝ(𝕋_{+}, M_{n}(ℝ)) and h ∈ C_{rd}(𝕋_{+}, ℝ^{n}), then for each (τ, η) ∈

𝕋_{+} × ℝ^{n} the initial value problem

x^{Δ} = A(t)x + h(t), x(τ) = η

has a unique solution x : 𝕋_{(τ)} → ℝ^{n}. □

A matrix X_{A} ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)) is said to be a matrix solution of (2.1) if each column of X_{A} satisfies (2.1). A fundamental matrix of (2.1) is a matrix solution X_{A} of (2.1) such that det X_{A}(t) ≠ 0 for all t ∈ 𝕋_{+}. A transition matrix of (2.1) at initial time τ ∈ 𝕋_{+} is a fundamental matrix such that X_{A}(τ) = I. The transition matrix of (2.1) at initial time τ ∈ 𝕋_{+} will be denoted by Φ_{A}(t, τ). Therefore, the transition matrix of (2.1) at initial time τ ∈ 𝕋_{+} is the unique solution of the following matrix initial value problem

Y^{Δ} = A(t)Y, Y (τ) = I,

and x(t) = Φ_{A}(t, τ)η, t ≥ τ, is the unique solution of initial value problem

x^{Δ} = A(t)x, x(τ) = η.

Theorem 2.2. ([15, Theorem 5.21]) If A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)) then

(i) Φ_{A}(t, t) = I;

(ii) Φ_{A}(σ(t), s) = [I + μ(t)A(t)]Φ_{A}(t, s);

(iii) Φ_{A}^{−1}(t, s) = Φ_{⊖A^{T}}^{T} (t, s);

(iv) Φ_{A}(t, s) = Φ_{A}^{−1}(s, t) = Φ_{⊖A^{T}}^{T} (s, t);

(v) Φ_{A}(t, s)Φ_{A}(s, r) = Φ_{A}(t, r), t ≥ s ≥ r. □

Theorem 2.3. ( [15, Theorem 5.24]) If A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)) and h ∈ C_{rd}(𝕋_{+}, ℝ^{n}), then for each (τ, η) ∈ 𝕋_{+} × ℝ^{n} the initial value problem

x^{Δ} = A(t)x + h(t), x(τ) = η

has a unique solution x : 𝕋_{(τ)} → ℝ^{n} given by

x(t) = Φ_{A}(t, τ)η + ∫_{τ}^{t} Φ_{A}(t, σ(s))h(s)Δs, t ≥ τ. □

As in the scalar case, along with (2.1), we consider its adjoint equation

y^{Δ} = −A^{T} (t)x^{σ}.  (2.3)

If A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)), then the initial value problem y^{Δ} = −A^{T} (t)x^{σ}, x(τ) = η, has a unique solution y : 𝕋_{(τ)} → ℝ^{n} given by y(t) = Φ_{⊖A^{T}} (t, τ)η, t ≥ τ.

Theorem 2.4. ( [15, Theorem 5.27]) If A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)) and h ∈ C_{rd}(𝕋_{+}, ℝ^{n}), then for each (τ, η) ∈ 𝕋_{+} × ℝ^{n} the initial value problem

y^{Δ} = −A^{T} (t)y^{σ} + h(t), x(τ) = η

has a unique solution x : 𝕋_{(τ)} → ℝ^{n} given by

y(t) = Φ_{⊖A^{T}} (t, τ)η + ∫_{τ}^{t} Φ_{⊖A}^{T}(t, s)h(s)Δs, t ∈ 𝕋_{(τ)}.

Lemma 2.1. Let τ ∈ 𝕋_{+}, y, b ∈ C_{rd}ℝ(𝕋_{+}, ℝ), p ∈ C_{rd}^{+}ℝ(𝕋_{+}, ℝ) and c, b_{k} ∈ ℝ_{+},

k = 1, 2, .... Then

y(t) ≤ c + ∫_{τ}^{t} p(s)y(s)Δs + ∑_{τ<t_{k}<t} b_{k}y(t_{k}), t ∈ 𝕋_{(τ)}  (2.4)

implies

y(t) ≤ c ∏_{τ<t_{k}<t}(1 + b_{k})e_{p}(t, τ), t ≥ τ.  (2.5)

Proof. Let v(t) := c + ∫_{τ}^{t} p(s)y(s)Δs + ∑_{τ<t_{k}<t} b_{k}y(t_{k}), t ≥ τ. Then

\{ \begin{matrix}v^{Δ}(t) = p(t)y(t), t ≠ t_{k}, v(τ) = c \\ v(t_{k}^{+}) = v(t_{k}) + b_{k}y(t_{k}), k = 1, 2, .... \end{matrix}

Since y(t) ≤ v(t), t ≥ τ, we then have

\{ \begin{matrix}v^{Δ}(t) ≤ p(t)v(t), t ≠ t_{k}, v(τ) = c \\ v(t_{k}^{+}) = (1 + b_{k})v(t_{k}), k = 1, 2, .... \end{matrix}

An application of Lemma A.4 yields

v(t) ≤ c ∏_{τ<t_{k}<t}(1 + b_{k})e_{p}(t, τ), t ≥ τ,

which implies (2.5). □

3 Homogeneous linear impulsive dynamic system on time scales

Consider the following homogeneous linear impulsive dynamic system on time scales

\{ \begin{matrix}x^{Δ} = A(t)x, t ∈ 𝕋_{+}, t ≠ t_{k} \\ x(t_{k}^{+}) = x(t_{k}^{−}) + B_{k}x(t_{k}), k = 1, 2, ... \end{matrix}  (3.1)

where B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)), 0 = t_{0} < t_{1} < t_{2} < ... < t_{k} < ..., with lim_{k→∞}t_{k} = ∞, x(t_{k}^{−}) represent the left limit of x(t) at t = t_{k}

(with x(t_{k}^{−}) = x(t) if t_{k} is left-scattered) and x(t_{k}^{+}) represents the right limit of x(t) at t = t_{k} (with x(t_{k}^{+}) = x(t) if t_{k} is right-scattered). We assume for the remainder of the paper that, for k = 1, 2, ..., the points of impulse t_{k} are rigth-dense.

Along with (3.1) we consider the following initial value problem

\{ \begin{matrix}x^{Δ} = A(t)x, t ∈ 𝕋_{(τ)}, t ≠ t_{k} \\ x(t_{k}^{+}) = x(t_{k}^{−}) + B_{k}x(t_{k}), k = 1, 2, ... \\ x(τ^{+}) = η, τ ≥ 0. \end{matrix}  (3.2)

We note that, instead of the usual initial condition x(τ) = η, we impose the limiting condition x(τ^{+}) = η which, in general case, is natural for (3.2) since (τ, η) may be such that τ = t_{k} for some k = 1, 2, .... In the case when τ ≠ t_{k} for any k, we shall understand the initial condition x(τ^{+}) = η in the usual sense, that is, x(τ) = η.

In order to define the solution of (3.2), we introduce the following spaces

Ω := {x : 𝕋_{+} → ℝ^{n}; x ∈ C((t_{k}, t_{k+1}), ℝ^{n}), k = 0, 1, ..., x(t_{k}^{+})
and x(t_{k}^{−}) exist with x(t_{k}^{−}) = x(t_{k}), k = 1, 2, ...}

and

Ω^{(1)} := {x ∈ Ω; x ∈ C^{1}((t_{k}, t_{k+1}), ℝ^{n}), k = 0, 1, ...},

where C((t_{k}, t_{k+1}),R^{n}) is the set of all continuous functions on (t_{k}, t_{k+1}) and C^{1}((t_{k}, t_{k+1}) is the set of all continuously differentiable functions on (t_{k}, t_{k+1}), k = 0, 1, ....

A function x ∈ Ω^{(1)} is said to be a solution of (3.1), if it satisfies x^{Δ}(t) = A(t)x(t), everywhere on 𝕋_{(τ)} \{τ, t_{k(τ)}, t_{k(τ)+1}, ...} and for each j = k(τ), k(τ)+ 1, ... satisfies the impulsive conditions x(t_{j}^{+}) = x(t_{j}) + B_{j}x(t_{j}) and the initial condition x(τ^{+}) = η, where k(τ) := min{k = 1, 2, ...; τ < t_{k}}.

Theorem 3.1. If A ∈ C_{rd}ℝ(𝕋_{+}, M_{n}(ℝ)) and B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., then any solution of (3.2) is also a solution of the impulsive integral equation

x(t) = x(τ) + ∫_{τ}^{t} A(s)x(s)Δs + ∑_{τ<t_{j}≤t} B_{j}x(t_{j}), t ≥ τ.  (3.3)

and conversely.

Proof. There exists i ∈ {1, 2, ...} such that τ ∈ [t_{i−1}, t_{i}). Then any solution of (3.2) on [τ, t_{i}) is also a solution of integral equation x(t) = x(τ) + ∫_{τ}^{t} A(s)x(s)Δs, t ∈ [τ, t_{i}]. Further, any solution of the initial value problem

\{ \begin{matrix}x^{Δ} = A(t)x, t ∈ (t_{i}, t_{i+1}) \\ x(t_{i}^{+}) = x(t_{i}) + B_{i}x(t_{i}). \end{matrix}

is a solution of integral equation x(t) = x(t_{i}^{+}) + ∫_{t_{i}}^{t} A(s)x(s)Δs, t ∈ [t_{i}, t_{i+1}). It follows that

x(t) = x(t_{i}^{+}) + ∫_{t_{i}}^{t} A(s)x(s)Δs = x(t_{i}) + B_{i}x(t_{i}) + ∫_{t_{i}}^{t} A(s)x(s)Δs
= x(τ) + ∫_{τ}^{t_{i}} A(s)x(s)Δs + ∫_{t_{i}}^{t} A(s)x(s)Δs + B_{i}x(t_{i})
= x(τ) + ∫_{τ}^{t} A(s)x(s)Δs + B_{i}x(t_{i}), t ∈ [t_{i}, t_{i+1}).

Next, we suppose that, for any k > i + 2, any solution of (3.2) on [t_{k−1}, t_{k}) is a solution of (3.3). Then any solution of the initial value problem

\{ \begin{matrix}x^{Δ} = A(t)x, t ∈ (t_{k}, t_{k+1}] \\ x(t_{k}^{+}) = x(t_{k}) + B_{k}x(t_{k}) \end{matrix}

is a solution of integral equation x(t) = x(t_{k}^{+}) + ∫_{t_{k}}^{t} A(s)x(s)Δs, t ∈ [t_{k}, t_{k+1}). It follows that

x(t) = x(t_{k}^{+}) + ∫_{t_{k}}^{t} A(s)x(s)Δs = x(t_{k}) + B_{k}x(t_{k}) + ∫_{t_{k}}^{t} A(s)x(s)Δs
= x(τ) + ∫_{τ}^{t_{k}} A(s)x(s)Δs + ∑_{τ<t_{j}<t_{k}} B_{j}x(t_{j}) + B_{k}x(t_{k}) + ∫_{t_{k}}^{t} A(s)x(s)Δs

= x(τ) + ∫_{τ}^{t} A(s)x(s)Δs + ∑_{τ<t_{j}≤t} B_{j}x(t_{j}), t ∈ [t_{k}, t_{k+1}).

Therefore, by the Mathematical Induction Principle, (3.3) is proved. The converse statement follows trivially and the proof is complete. □

Theorem 3.2. If A ∈ C_{rd}ℝ(𝕋_{+}, M_{n}(ℝ)) and B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., then the solution of (3.2) satisfies the following estimate

||x(t)|| ≤ ||x(τ)|| ∏_{τ<t_{k}≤t}(1 + ||B_{k}||) exp (∫_{τ}^{t} ||A(s)||Δs),  (3.4)

for τ, t ∈ 𝕋_{+} with t ≥ τ.

Proof. From (3.3) we obtain that

||x(t)|| ≤ ||x(τ)|| + ∫_{τ}^{t} ||A(s)|| · ||x(s)||Δs + ∑_{τ<t_{j}≤t} ||B_{j}|| · ||x(t_{j})||, t ≥ τ.

Then, by Lemma 2.1, it follows that

||x(t)|| ≤ ||x(τ)|| ∏_{τ<t_{k}≤t}(1 + ||B_{k}||) exp_{||A(·)||}(t, τ), t ≥ τ.

Since for any a ≥ 0,

lim_{u↘μ(s)} \frac{ln(1 + au)}{u} = \{ \begin{matrix}a & if μ(s) = 0 \\ \frac{ln(1+aμ(s))}{μ(s)} ≤ a & if μ(s) > 0, \end{matrix}

then explicit estimation of the modulus of the exponential function on time scales (see [24]) gives

exp_{||A(·)||}(t, τ) = exp (∫_{τ}^{t} lim_{u↘μ(s)} \frac{ln(1 + u||A(s)||)}{u}Δs)
≤ exp (∫_{τ}^{t} ||A(s)||Δs), t ≥ τ.

Thus we obtain (3.4). □

Along with (3.1) we consider the impulsive transition matrix S_{A}(t, s), 0 ≤ s ≤ t, associated with {B_{k}, t_{k}}_{k=1}^{∞}, given by

S_{A}(t, s)= \{ \begin{matrix}Φ_{A}(t, s) if t_{k−1} ≤ s ≤ t ≤ t_{k}; \\ Φ_{A}(t, t_{k}^{+})(I + B_{k})Φ_{A}(t_{k}, s) if t_{k−1} ≤ s < t_{k} < t < t_{k+1}; \\ Φ_{A}(t, t_{k}^{+})[ ∏_{s<t_{j}≤t}(I + B_{j})Φ_{A}(t_{j}, t_{j−1}^{+})](I + B_{i})Φ_{A}(t_{i}, s) \\ if t_{i−1} ≤ s < t_{i} < ... < t_{k} < t < t_{k+1}, \end{matrix}

(3.5)

where Φ_{A}(t, s), 0 ≤ s ≤ t, is the transition matrix of (2.1) at initial time s ∈ 𝕋_{+}.

Remark 3.1. Since

\begin{matrix}S_{A}(t, s) = \\ Φ_{A}(t, t_{k}^{+})(I + B_{k})Φ_{A}(t_{k}, t_{k−1}^{+})[ ∏_{s<t_{j}<t_{k}} (I + B_{j})Φ_{A}(t_{j}, t_{j−1}^{+})](I + B_{i})Φ_{A}(t_{i}, s), \end{matrix}

it follows that

S_{A}(t, s) =Φ_{A}(t, t_{k}^{+})(I + B_{k})S_{A}(t_{k}, s) for t_{i−1} ≤ s < t_{i} < ... < t_{k} ≤ t < t_{k+1}.

In the following, we will assume that I +B_{k} is invertible for each k = 1, 2, ....

Theorem 3.3. If A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)) and B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., then for each (τ, η) ∈ 𝕋_{+} × ℝ^{n} the initial value problem (3.2) has a unique solution given by

x(t) = S_{A}(t, τ)η, t ≥ τ.

Proof. Let (τ, η) ∈ 𝕋_{+} × ℝ^{n}. Then there exists i ∈ {1, 2, ...} such that τ ∈ [t_{i−1}, t_{i}). Then the unique solution of (3.2) on [τ, t_{i}) is given by x(t) = Φ_{A}(t, τ)η = S_{A}(t, τ)η, t ∈ [τ, t_{i}).

Further, we consider the initial value problem

\{ \begin{matrix}x^{Δ} = A(t)x, t ∈ (t_{i}, t_{i+1}) \\ x(t_{i}^{+}) = x(t_{i}) + B_{i}x(t_{i}). \end{matrix}

This initial value problem has the unique solution given by x(t) = Φ_{A}(t, t_{i}^{+})x(t_{i}^{+}), t ∈ [t_{i}, t_{i+1}). It follows that

x(t) = Φ_{A}(t, t_{i}^{+})x(t_{i}^{+}) = Φ_{A}(t, t_{i}^{+})(I + B_{i})x(t_{i})
= Φ_{A}(t, t_{i}^{+})(I + B_{i})Φ_{A}(t_{i}, τ)η
S_{A}(t, τ)η,

and so x(t) = S_{A}(t, τ)η, t ∈ [t_{i}, t_{i+1}). Next, we suppose that, for any k > i + 2, the unique solution of (3.2) on [t_{k−1}, t_{k}] is given by

x(t) = S_{A}(t, τ)η = Φ_{A}(t, t_{k−1}^{+})[ ∏_{s<t_{j}<t}(I + B_{j})Φ_{A}(t_{j}, t_{j−1}^{+})](I + B_{i})Φ_{A}(t_{i}, τ)η.

Then the initial value problem

\{ \begin{matrix}x^{Δ} = A(t)x, t ∈ (t_{k}, t_{k+1}) \\ x(t_{k}^{+}) = x(t_{k}) + B_{k}x(t_{k}) \end{matrix}

has the unique solution x(t) = Φ_{A}(t, t_{k}^{+})x(t_{k}^{+}), t ∈ [t_{k}, t_{k+1}). It follows that

x(t) = Φ_{A}(t, t_{k}^{+})x(t_{i}^{+}) = Φ_{A}(t, t_{k}^{+})(I + B_{k})x(t_{k}^{+}) =

Φ_{A}(t, t_{k}^{+})(I + B_{k})Φ_{A}(t, t_{k−1}^{+})[ ∏_{s<t_{j}≤t} (I + B_{j})Φ_{A}(t_{j}, t_{j−1}^{+})](I + B_{i})Φ_{A}(t_{i}, τ)η

= Φ_{A}(t, t_{k}^{+})[ ∏_{s<t_{j}<t} (I + B_{j})Φ_{A}(t_{j}, t_{j−1}^{+})](I + B_{i})Φ_{A}(t_{i}, τ)

= S_{A}(t, τ)η,

and so x(t) = S_{A}(t, τ)η, t ∈ [t_{k}, t_{k+1}]. Therefore, by the Mathematical Induction Principle, the theorem is proved. □

Corollary 3.1. If A ∈ C_{rd}ℝ(𝕋_{+}, M_{n}(ℝ)) and B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., then the impulsive transition matrix S_{A}(t, s), 0 ≤ s ≤ t, is the unique solution of the following matrix initial value problem

\{ \begin{matrix}Y^{Δ} = A(t)Y, t ∈ 𝕋_{(s)}, t ≠ t_{k} \\ Y (t_{k}^{+}) = (I + B_{k})Y (t_{k}), k = 1, 2, ... \\ Y (s^{+}) = I, s ≥ 0. \end{matrix}  (3.6)

Moreover, the following properties hold:

(i) S_{A}(t_{k}^{+}, s) = (I + B_{k})S_{A}(t_{k}, s), t_{k} ≥ s, k = 1, 2, ...; (ii) S_{A}(t, t_{k}^{+}) = S_{A}(t, t_{k})(I + B_{k})^{−1}, t_{k} ≤ t, k = 1, 2, ...; (iii) S_{A}(t, t_{k}^{+})S_{A}(t_{k}^{+}, s) = S_{A}(t, s), 0 ≤ s ≤ t_{k} ≤ t, k = 1, 2, .... □

From the Theorems 3.2 and 3.3, we obtain the following result.

Corollary 3.2. If A ∈ C_{rd}ℝ(𝕋_{+}, M_{n}(ℝ)) and B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., then we have the following estimate

||S_{A}(t, τ)|| ≤ ∏_{τ<t_{k}≤t}(1 + ||B_{k}||) exp (∫_{τ}^{t} ||A(s)||Δs),  (3.7)

for τ, t ∈ 𝕋_{+} with t ≥ τ. Moreover, for any τ, t ∈ 𝕋_{+} the function (r, s) → S_{A}(r, s) is bounded on set {(r, s) ∈ 𝕋_{+} × 𝕋_{+}; τ ≤ s ≤ r ≤ t}.

Proof. Using the Theorems 3.2 and 3.3, for all τ, t ∈ 𝕋_{+} with t ∈ 𝕋_{(τ)}, it follows

||x(t)|| = ||S_{A}(t, τ)x(τ)|| ≤ ||x(τ)|| ∏_{τ<t_{k}≤t}(1 + ||B_{k}||) exp (∫_{τ}^{t} ||A(s)||Δs),

which implies (3.7). □

Let X_{A}(t), t ∈ 𝕋_{+}, be the unique solution of (3.6) with the initial condition Y (0) = I, i.e., X_{A}(t) := S_{A}(t, 0), t ∈ 𝕋_{+}.

Theorem 3.4. If A ∈ C_{rd}ℝ(𝕋_{+}, M_{n}(ℝ)) and B_{k} ∈ M_{n}(ℝ), then the impulsive transition matrix S_{A}(t, s) has the following properties

(i) S_{A}(t, s) = X_{A}(t)X_{A}^{−1}(s), 0 ≤ s ≤ t; (ii) S_{A}(t, t) = I, t ≥ 0; (iii) S_{A}(t, s) = S_{A}^{−1}(s, t), 0 ≤ s ≤ t; (iv) S_{A}(σ(t), s) = [I + μ(t)A(t)]S_{A}(t, s), 0 ≤ s ≤ t;

(v) S_{A}(t, s)S_{A}(s, r) = S_{A}(t, r), 0 ≤ r ≤ s ≤ t.

Proof. (i). Let Y (t) := X_{A}(t)X_{A}^{−1}(s), 0 ≤ s ≤ t. Then we have that

Y^{Δ}(t) = X_{A}^{Δ}(t)X_{A}^{−1}(s) = A(t)X_{A}(t)X_{A}^{−1}(s) = A(t)Y (t), t ≠ t_{k}. Also, Y (s) = X_{A}(s)X_{A}^{−1}(s) = I, and

Y (t_{k}^{+}) − Y (t_{k}) = X_{A}(t_{k}^{+})X_{A}^{−1}(s) − X_{A}(t_{k})X_{A}^{−1}(s)

= [X_{A}(t_{k}^{+}) − X_{A}(t_{k})]X_{A}^{−1}(s) = B_{k}X_{A}(t_{k})X_{A}^{−1}(s)
= B_{k}Y (t_{k})

EJQTDE, 2010 No. 11, p. 9 for each t_{k} ≥ s. Therefore, Y (t) = X_{A}(t)X_{A}^{−1}(s) solves the initial value problem (3.6), which has exactly one solution. Therefore, S_{A}(t, s) = X_{A}(t)X_{A}^{−1}(s), 0 ≤ s ≤ t.

The properties (ii) and (iii) follows from (i). Now, from Theorem A.2 and Corollary 3.1, we have that

S_{A}(σ(t), s) = S_{A}(t, s) + μ(t)S_{A}^{Δ}(t, s)
= S_{A}(t, s) + μ(t)A(t)S_{A}(t, s)
= [I + μ(t)A(t)]S_{A}(t, s),

and so (iv) is true.

Further, let Y (t) := S_{A}(t, s)S_{A}(s, r), 0 ≤ r ≤ s ≤ t. Then we have

Y^{Δ}(t) = S_{A}^{Δ}(t, s)S_{A}(s, r) = A(t)S_{A}(t, s)S_{A}(s, r) = A(t)Y (t), t ≠ t_{k},

and Y (r^{+}) = S_{A}(r^{+}, s)S_{A}(s, r^{+}) = S_{A}(r^{+}, s)S_{A}^{−1}(r^{+}, s) = I according to (iii). Also,

\begin{matrix}Y (t_{k}^{+}) = S_{A}(t_{k}^{+}, s)S_{A}(s, r) = (I + B_{k})S_{A}(t_{k}, s)S_{A}(s, r) \\ = (I + B_{k})S_{A}(t_{k}, r) = (I + B_{k})Y (t_{k}) \end{matrix}

for each t_{k} ≥ s. Therefore, Y (t) solves (3.6) with initial condition Y (r^{+}) = I, r ∈ 𝕋_{+}. By the uniqueness of solution, it follows that S_{A}(t, r) = Y (t) = S_{A}(t, s)S_{A}(s, r), 0 ≤ r ≤ s ≤ t, and so (v) is true. □

Theorem 3.5. If A ∈ C_{rd}ℝ(𝕋_{+}, M_{n}(ℝ)), B_{k} ∈ M_{n}(ℝ) and a, b, τ ∈ 𝕋_{+}, then

\frac{∂}{Δs}S_{A}(t, s) = −S_{A}(t, σ(s))A(s), s ≠ t_{k}.

Proof. Indeed, from Theorem 3.4 and Theorem A.2, we have

\frac{∂}{Δt}S_{A}(t, s) = \frac{∂}{Δt}S_{A}^{−1}(s, t) = −S_{A}^{−1}(σ(s), t)[\frac{∂}{Δt}S_{A}(s, t)]S_{A}^{−1}(s, t)
= −S_{A}^{−1}(σ(s), t)A(s)S_{A}(s, t)S_{A}^{−1}(s, t)
= −S_{A}(t, σ(s))A(s).

Therefore, \frac{∂}{Δt}S_{A}(t, s) = −S_{A}(t, σ(s))A(s) for all s ∈ 𝕋_{+} , s ≠ t_{k}, k = 1, 2, .... □

Theorem 3.6. If A ∈ C_{rd}ℝ(𝕋_{+}, M_{n}(ℝ)) and B_{k} ∈ M_{n}(ℝ), then

S_{A}^{−1}(t, s) = S_{⊖A^{T}}^{T} (t, s), 0 ≤ s ≤ t.  (3.8)

Proof. Let Y (t) := (S_{A}^{−1}(t, s))^{T}, 0 ≤ s ≤ t. According to Theorem A.2 and (6.5), we have

Y^{Δ}(t) = − (S_{A}^{−1}(σ(t), s)S_{A}^{Δ}(t, s)S_{A}^{−1}(t, s))^{T}
= − (S_{A}^{−1}(σ(t), s)A(t)S_{A}(t, s)S_{A}^{−1}(t, s))^{T}
= − (S_{A}^{−1}(σ(t), s)A(t))^{T}
= −(S_{A}^{−1}(t, s)[I + μ(t)A(t)]^{−1}A(t))^{T}
= (S_{A}^{−1}(t, s)(⊖A)(t))^{T} = (⊖A)^{T} (t))(S_{A}^{−1}(t, s))^{T}
= (⊖A)^{T} (t)Y (t),

and hence

Y^{Δ} = (⊖A)^{T} (t)Y, t ≥ s, t ≠ t_{k}.

Also, Y (s^{+}) = (S_{A}^{−1}(s^{+}, s^{+}))^{T} = (I^{−1})^{T} = I and

Y (t_{k}^{+}) = (S_{A}^{−1}(t_{k}^{+}, s))^{T} = (X_{A}(s)X_{A}^{−1}(t_{k}^{+}))^{T} = ((X_{A}^{T})^{−1}(s)X_{A}^{T}(t_{k}^{+}))^{−1}

= ((X_{A}^{T})^{−1}(s)X_{A}^{T}(t_{k})(I + B_{k}^{T}))^{−1} = (I + B_{k}^{T})^{−1}(X_{A}^{T})^{−1}(t_{k})X_{A}^{T}(s)

= (I + B_{k}^{T})^{−1}(X_{A}(s)X_{A}^{−1}(t_{k}))^{T} = (I + B_{k}^{T})^{−1}(S_{A}^{−1}(t_{k}, s))^{T}

= (I + C_{k})Y (t_{k})

for each t_{k} ≥ s, where C_{k} := −B^{T} (I + B_{k}^{T})^{−1}, k = 1, 2, .... Therefore, Y (t) = (S_{A}^{−1}(t, s))^{T}, 0 ≤ s ≤ t, solve the initial value problem

\{ \begin{matrix}Y^{Δ} = (⊖A)^{T} (t)Y, t ≥ s, t ≠ t_{k}. \\ Y (t_{k}^{+}) = (I + C_{k})Y (t_{k}), k = 1, 2, ... \\ Y (s^{+}) = I, \end{matrix}

which has exactly one solution. It follows that S_{⊖A^{T}} (t, s) = Y (t) = (S_{A}^{−1}(t, s))^{T}, and so S_{A}^{−1}(t, s) = S_{⊖A^{T}}^{T} (t, s), 0 ≤ s ≤ t. □

Remark 3.2. The matrix equation Y^{Δ} = −A^{T} (t)Y^{σ} is equivalent to the equation Y^{Δ} = (⊖A)^{T} (t)Y.

Indeed, we have

y^{Δ} = −A^{T} (t)y^{σ} = −A^{T} (t)[y + μ(t)y^{Δ}]
= −A^{T} (t)y − μ(t)A^{T} (t)y^{Δ},

that is,

[I + μ(t)A^{T} (t)]y^{Δ} = −A^{T} (t)y.

Since A is regressive, then A^{T} is also regressive. Then, by (6.5), the above equality is equivalent to

y^{Δ} = −[I + μ(t)A^{T} (t)]^{−1}A^{T} (t)y = (⊖A^{T})(t)y.

Remark 3.3. If A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)) and B_{k} ∈ M_{n}(ℝ), then y(t) = S_{⊖A^{T}}^{T} (t, τ)η, 0 ≤ τ ≤ t, is the unique solution of initial value problem

\{ \begin{matrix}y^{Δ} = −A^{T} (t)y^{σ}, t ≥ τ, t ≠ t_{k}. \\ y(t_{k}^{+}) = (I + C_{k})y(t_{k}), k = 1, 2, ... \\ y(τ^{+}) = η, \end{matrix}  (3.9)

where C_{k} := −B^{T} (I + B_{k}^{T})^{−1}, k = 1, 2, ....

The homogeneous linear impulsive dynamic system on time scales (3.9) is called the adjoint dynamic system of (3.2).

Corollary 3.3. If A ∈ C_{rd}R(T_{+}, M_{n}(ℝ)) and B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., then any solution of (3.9) is also a solution of the impulsive integral equation

y(t) = y(τ) − ∫_{τ}^{t} A^{T} (s)y^{σ}(s)Δs + ∑_{τ<t_{j}<t} B_{j}^{T} y(t_{j}), t ∈ 𝕋_{(τ)}.

and conversely.

4 Nonhomogeneous linear impulsive dynamic system on time scales

Let l^{∞}(ℝ^{n}) be the space of all sequences c := {c_{k}}_{k=1}^{∞}, c_{k} ∈ ℝ^{n}, k = 1, 2, ...,such that sup_{k≥1} ||c_{k}|| < ∞. Then l^{∞}(ℝ^{n}) is a Banach space with the norm ||c|| := sup_{k≥1} ||c_{k}||.

Consider the following nonhomogeneous initial value problem

\{ \begin{matrix}x^{Δ} = A(t)x + h(t), t ∈ 𝕋_{(τ)}, t ≠ t_{k} \\ x(t_{k}^{+}) = x(t_{k}) + B_{k}x(t_{k}) + c_{k}, k = 1, 2, ... \\ x(τ^{+}) = η, τ ≥ 0. \end{matrix}  (4.1)

where B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)), c := {c_{k}}_{k=1}^{∞} ∈ l^{∞}(ℝ^{n}) and h : 𝕋_{+} → ℝ^{n} is a given function.

Theorem 4.1. If B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)), c := {c_{k}}_{k=1}^{∞} ∈ l^{∞}(ℝ^{n}) and h ∈ C_{rd}R(𝕋_{+}, ℝ^{n}) then, for each (τ, η) ∈ 𝕋_{+} × ℝ^{n}, the initial value problem (4.1) has a unique solution given by

x(t) = S_{A}(t, τ)η + ∫_{τ}^{t} S_{A}(t, σ(s))h(s)Δs + ∑_{τ<t_{j}<t} S_{A}(t, t_{j}^{+})c_{j}, t ≥ τ. (4.2)

Proof. Let (τ, η) ∈ 𝕋_{+} × ℝ^{n}. Then there exists i ∈ {1, 2, ...} such that τ ∈ [t_{i−1}, t_{i}). Then the unique solution of (4.1) on [τ, t_{i}] is given by

x(t) = Φ_{A}(t, τ)η + ∫_{τ}^{t} Φ_{A}(t, σ(s))h(s)Δs
= S_{A}(t, τ)η + ∫_{τ}^{t} S_{A}(t, σ(s))h(s)Δs, t ∈ [τ, t_{i}].

For t ∈ (t_{i}, t_{i+1}] the Cauchy problem

\{ \begin{matrix}x^{Δ} = A(t)x + h(t), t ∈ (t_{i}, t_{i+1}) \\ x(t_{i}^{+}) = x(t_{i}) + B_{i}x(t_{i}) + c_{i}, \end{matrix}

has the unique solution

x(t) = Φ_{A}(t, t_{i}^{+})x(t_{i}^{+}) + ∫_{t_{i}}^{t} Φ_{A}(t, σ(s))h(s)Δs, t ∈ (t_{i}, t_{i+1}]. It follows that x(t)= Φ_{A}(t, t_{i}^{+})[(I + B_{i})x(t_{i}) + c_{i}]+ ∫_{t_{i}}^{t} Φ_{A}(t, σ(s))h(s)Δs =Φ_{A}(t, t_{i}^{+})(I + B_{i})[Φ_{A}(t_{i}, τ)η+ ∫_{τ}^{t_{i}} S_{A}(t_{i}, σ(s))h(s)Δs] +Φ_{A}(t, t_{i}^{+})c_{i}+ ∫_{t_{i}}^{t} Φ_{A}(t, σ(s))h(s)Δs = Φ_{A}(t, t_{i}^{+})(I + B_{i})Φ_{A}(t_{i}, τ)η+ ∫_{τ}^{t_{i}} Φ_{A}(t, t_{i}^{+})(I + B_{i})Φ_{A}(t_{i}, σ(s))h(s)Δs +Φ_{A}(t, t_{i}^{+})c_{i}+ ∫_{t_{i}}^{t} Φ_{A}(t, σ(s))h(s)Δs Using (3.5) we get that

\begin{matrix}x(t) = \\ S_{A}(t, τ)η + ∫_{τ}^{t_{k}} S_{A}(t, σ(s))h(s)Δs + ∫_{t_{k}}^{t} S_{A}(t, σ(s))h(s)Δs + S_{A}(t, t_{i}^{+})c_{i} \end{matrix}

and so

x(t) = S_{A}(t, τ)η + ∫_{τ}^{t} S_{A}(t, σ(s))h(s)Δs + S_{A}(t, t_{i}^{+})c_{i}, t ∈ (t_{i}, t_{i+1}]. Next, we suppose that, for any k > i + 2, the unique solution of (4.1) on

[t_{k−1}, t_{k}] is given by

x(t) = S_{A}(t, τ)η + ∫_{τ}^{t} S_{A}(t, σ(s))h(s)Δs + ∑_{τ<t_{j}<t} S_{A}(t, t_{j}^{+})c_{j}, t ∈ [t_{k}, t_{k+1}].

Then the initial value problem

\{ \begin{matrix}x^{Δ} = A(t)x + h(t), t ∈ (t_{k}, t_{k+1}] \\ x(t_{k}^{+}) = x(t_{k}) + B_{k}x(t_{k}) + c_{k} \end{matrix}

has the unique solution

x(t) = Φ_{A}(t, t_{k}^{+})x(t_{k}^{+}) + ∫_{t_{k}}^{t} Φ_{A}(t, σ(s))h(s)Δs, t ∈ [t_{k}, t_{k+1}].

It follows that EJQTDE, 2010 No. 11, p. 13 x(t)= Φ_{A}(t, t_{k}^{+})[(I + B_{k})x(t_{k}) + c_{k}]+ ∫_{t_{k}}^{t} Φ_{A}(t, σ(s))h(s)Δs = Φ_{A}(t, t_{k}^{+})(I + B_{k})[S_{A}(t_{k}, τ)η+ ∫_{τ}^{t_{k}} S_{A}(t_{k}, σ(s))h(s)Δs+ ∑_{τ<t_{j}<t_{k}} S_{A}(t_{k}, t_{j}^{+})c_{j}] +Φ_{A}(t, t_{k}^{+})c_{k}+ ∫_{t_{k}}^{t} Φ_{A}(t, σ(s))h(s)Δs, hence

x(t)=

Φ_{A}(t, t_{k}^{+})(I + B_{k})S_{A}(t_{k}, τ)η+ ∫_{τ}^{t_{k}} Φ_{A}(t, t_{k}^{+})(I + B_{k})S_{A}(t_{k}, σ(s))h(s)Δs + ∑_{τ<t_{j}<t_{k}} Φ_{A}(t, t_{k}^{+})(I + B_{k})S_{A}(t_{k}, t_{j}^{+})c_{j}+S_{A}(t, t_{k}^{+})c_{k} + ∫_{t_{k}}^{t} S_{A}(t, σ(s))h(s)Δs. Using the Remark 3.1, we obtain that

x(t) = S_{A}(t, τ)η + ∫_{τ}^{t_{k}} S_{A}(t, σ(s))h(s)Δs + ∫_{t_{k}}^{t} S_{A}(t, σ(s))h(s)Δs

+ ∑_{τ<t_{j}<t} S_{A}(t, t_{j}^{+})c_{j},

and so,

x(t) = S_{A}(t, τ)η + ∫_{τ}^{t} S_{A}(t, σ(s))h(s)Δs + ∑_{τ<t_{j}<t} S_{A}(t, t_{j}^{+})c_{j},

Therefore, by the Mathematical Induction Principle, (4.2) is proved. □

Corollary 4.1. If B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)), and h ∈ C_{rd}R(𝕋_{+}, ℝ^{n}) then, for each (τ, η) ∈ 𝕋_{+} × ℝ^{n}, the initial value problem

\{ \begin{matrix}x^{Δ} = A(t)x + h(t), t ∈ 𝕋_{(τ)}, t ≠ t_{k} \\ x(t_{k}^{+}) = x(t_{k}) + B_{k}x(t_{k}), k = 1, 2, ... \\ x(τ^{+}) = η, τ ≥ 0. \end{matrix}

has a unique solution given by x(t) = S_{A}(t, τ)η + ∫_{τ}^{t} S_{A}(t, σ(s))h(s)Δs, t ∈ 𝕋_{(τ)}.

Theorem 4.2. If B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)), c := {c_{k}}_{k=1}^{∞} ∈ l^{∞}(ℝ^{n}) and h ∈ C_{rd}R(𝕋_{+}, ℝ^{n}) then, for each (τ, η) ∈ 𝕋_{+} × ℝ^{n}, the initial value problem

\{ \begin{matrix}y^{Δ} = −A^{T} (t)y^{σ} + h(t), t ≥ τ, t ≠ t_{k}. \\ y(t_{k}^{+}) = (I + C_{k})y(t_{k}) + c_{k}, k = 1, 2, ... \\ y(τ^{+}) = η. \end{matrix}  (4.3)

has a unique solution given by

y(t) = S_{⊖A^{T}} (t, τ)η + ∫_{τ}^{t} S_{⊖A^{T}} (t, s)h(s)Δs + ∑_{τ<t_{j}<t} S_{⊖A^{T}} (t, t_{j}^{+})c_{j}, t ≥ τ, (4.4)

where C_{k} := −B^{T} (I + B_{k}^{T})^{−1}, k = 1, 2, ....

Proof. We have

y^{Δ} = −A^{T} (t)y^{σ} + h(t) = −A^{T} (t)[y + μ(t)y^{Δ}] + h(t)
= −A^{T} (t)y − μ(t)A^{T} (t)y^{Δ} + h(t),

that is,

[I + μ(t)A^{T} (t)]y^{Δ} = −A^{T} (t)y + h(t).

Since A is regressive, then A^{T} is also regressive, and thus the above inequality is equivalent to

y^{Δ} = −[I + μ(t)A^{T} (t)]^{−1}A^{T} (t)y + [I + μ(t)A^{T} (t)]^{−1}h(t)
= (⊖A^{T})(t)y + [I + μ(t)A^{T} (t)]^{−1}h(t).

From Theorem 4.1 it follows that (4.3) has the unique solution

\begin{matrix}y(t) = S_{⊖A^{T}} (t, τ)η + ∫_{τ}^{t} S_{⊖A^{T}} (t, σ(s))[I + μ(s)A^{T} (s)]^{−1}h(s)Δs \\ + ∑_{τ<t_{j}<t} S_{⊖A^{T}} (t, t_{j}^{+})c_{j}, t ≥ τ. \end{matrix}  (4.5)

Since, by Theorem 3.4 and (3.8), we have

S_{⊖A^{T}} (t, σ(s))[I + μ(s)A^{T} (s)]^{−1} = {[I + μ(s)A^{T} (s)]^{−1}S_{⊖A^{T}}^{T} (t, σ(s))}^{T}

= {[I + μ(s)A^{T} (s)]^{−1}S_{A}^{−1}(t, σ(s))}^{T} = {[I + μ(s)A^{T} (s)]^{−1}S_{A}(σ(s), t)}^{T}

= S_{A}^{T}(s, t) = (S_{A}^{−1}(s, t))^{T} = S_{⊖A^{T}} (t, s),

then from (4.5) we obtain (4.4). □

5 Boundedness and stability of linear impulsive dynamic system on time scales

Definition 5.1. The dynamic system (3.2) is said to be exponentially stable (e.s.) if there exists a positive constant λ with −λ ∈ R^{+} such that for every τ ∈ 𝕋_{+}, there exists N = N(τ) ≥ 1 such that the solution of (3.2) through (τ, x(τ)) satisfies

||x(t)|| ≤ N||x(τ)||e_{−λ}(t, τ) for all t ∈ 𝕋_{(τ)}.

The dynamic system (3.2) is said to be uniformly exponentially stable (u.e.s.) if it is e.s. and the constant N can be chosen independently of τ ∈ 𝕋_{+}.

Theorem 5.1. Suppose that B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)), and there exists a positive constant θ such that t_{k+1} − t_{k} < θ, k = 1, 2, .... If the solution of initial value problem

\{ \begin{matrix}x^{Δ} = A(t)x, t ∈ 𝕋_{(τ)}, t ≠ t_{k} \\ x(t_{k}^{+}) = x(t_{k}) + B_{k}x(t_{k}) + c_{k}, k = 1, 2, ... \\ x(τ^{+}) = 0, τ ≥ 0, \end{matrix}  (5.1)

is bounded for any c := {c_{k}}_{k=1}^{∞} ∈ l^{∞}(ℝ^{n}), then there exists a positive constants N = N(τ) ≥ 1, λ with −λ ∈ R^{+} such that

||S_{A}(t, τ|| ≤ Ne_{−λ}(t, τ) for all t ∈ 𝕋_{(τ)}.

Proof. From Theorem 4.1, the solution of (5.1) is given by

x(t) = ∑_{τ<t_{j}<t} S_{A}(t, t_{j}^{+})c_{j}, t ∈ 𝕋_{(τ)}.  (5.2)

For each fixed t ∈ 𝕋_{(τ)}, by the Corollary 3.2, the operator U_{t} : l^{∞}(ℝ^{n}) → ℝ^{n}, given by U_{t}(c) := x(t), is a bounded linear operator. In fact, ||U_{t}(c)|| ≤ ∑_{τ<t_{j}<t} ||S_{A}(t, t_{j}^{+})|| · ||c||_{l}^{∞} < ∞ for any c ∈ l^{∞}(ℝ^{n}). Since the solution x(t) of (5.1) is bounded for any c ∈ l^{∞}(ℝ^{n}), then uniform boundedness principle implies that there exists a constant K > 0 such that

||x(t)|| ≤ K||c||_{l^{∞}} for all c ∈ l^{∞}(ℝ^{n}) and τ ∈ 𝕋_{+},

that is,

|| ∑_{τ<t_{j}<t} S_{A}(t, t_{j}^{+})c_{j}|| ≤ K||c||_{l}^{∞} for all c ∈ l^{∞}(ℝ^{n}) and τ ∈ 𝕋_{+}.  (5.3)

Let τ ∈ 𝕋_{+} be fixed. Then there exists i ∈ {1, 2, ...} such that τ ∈ [t_{i−1}, t_{i}). We define the sequences {β_{j}}_{j=1}^{∞} and {c_{j}}_{j=1}^{∞} ∈ l^{∞}(ℝ^{n}) given by

β_{j} := \{ \begin{matrix}0 & if j < i \\ ||S_{A}(t_{j}^{+}, τ)||, & if j ≥ i \end{matrix} and c_{j} := \{ \begin{matrix}0 & if j < i \\ \frac{1}{β_{j}} S_{A}(t_{j}^{+}, τ)y & if j ≥ i \end{matrix}

respectively. Here y is an arbitrary fixed element of ℝ^{n}\{0}. Also, we observe that ||c||_{l}^{∞} = 1. Hence, from (5.3) we obtain

||S_{A}(t, τ)y|| ∑_{τ<t_{j}<t} \frac{1}{β_{j}} ≤ K||y|| for all t ∈ 𝕋_{(τ)}.

Since y is an arbitrary element of ℝ^{n}\{0} then it follows that

||S_{A}(t, τ)|| ∑_{τ<t_{j}<t} \frac{1}{β_{j}} ≤ K for all t ∈ 𝕋_{(τ)}.  (5.4)

Now, if t_{k} > t_{i} and t ∈ [τ, t_{k+1}) then, from (5.4), we obtain that

||S_{A}(t_{k}^{+}, τ)|| ∑_{τ<t_{j}≤t_{k}} \frac{1}{β_{j}} ≤ K for all t ∈ [τ, t_{k+1}),

that is,

β_{k} ∑_{τ<t_{j}≤t_{k}} \frac{1}{β_{j}} ≤ K for all t ∈ [τ, t_{k+1}).  (5.5)

If t ∈ [τ, t_{i+1}) then, from (5.4), we obtain

||S_{A}(t, τ)|| ≤ Kβ_{i} for all t ∈ (t_{i}, t_{i+1}).

Without loss of generality we can assume that K > 1.

Further, if t ∈ [τ, t_{i+2}) then, from (5.5), we obtain β_{i+1}(\frac{1}{β_{i}} + \frac{1}{β_{i+1}} ) ≤ K, and so β_{i+1} ≤ (K − 1)β_{i}.

If t ∈ [τ, t_{i+3}) then, from (5.5), we obtain β_{i+2}(\frac{1}{β_{i}} + \frac{1}{β_{i+1}} + \frac{1}{β_{i+2}} ) ≤ K. Then

1 + \frac{β_{i+2}}{(K − 1)β_{i}} + \frac{β_{i+2}}{β_{i}} ≤ 1 + \frac{β_{i+2}}{β_{i+1}} + \frac{β_{i+2}}{β_{i}} ≤ K,

that is, \frac{β_{i+2}}{β_{i}} (\frac{1}{K−1} + 1) ≤ K − 1. It follows that β_{i+2} ≤ \frac{(K−1)^{2}}{K} β_{i}.

Further, we prove by induction that if t ∈ (τ, t_{i+j}) then

β_{i+j} ≤ \frac{(K − 1)^{j}}{K^{j−1}} β_{i} for all j ≥ 1.  (5.6)

Suppose that (5.6) is true for all j ≤ l − 1. If t ∈ [τ, t_{i+l}) then, from (5.5) we have that

β_{i+l}(\frac{1}{β_{i}} + \frac{1}{β_{i+1}} + ... + \frac{1}{β_{i+l}}) ≤ K.

It follows that

1 + β_{i+l}[\frac{1}{β_{i}} + \frac{1}{(K − 1)β_{i}} + \frac{K}{(K − 1)^{2}β_{i}} + ... + \frac{K^{l−2}}{(K − 1)^{l−1}β_{i}}]

≤ 1 + β_{i+l}(\frac{1}{β_{i}} + \frac{1}{β_{i+1}} + ... + \frac{1}{β_{i+l−1}} ) ≤ K,

hence

\frac{β_{i+l}}{β_{i}} [1 + \frac{1}{K − 1} + \frac{K}{(K − 1)^{2}} + ... + \frac{K^{l−2}}{(K − 1)^{l−1}} ] ≤ K − 1

and so, β_{i+l} ≤ \frac{(K−1)^{l}}{K^{l−1}} β_{i}. Now, let j ≥ 1 be fixed and let \{c_{l}\}_{l=1}^{∞} ∈ l^{∞}(ℝ^{n}) be the sequence defined by

c_{l} := \{ \begin{matrix}0 & if l ≠ i + j \\ S_{A}(t_{l}^{+}, τ)y & if l = i + j \end{matrix}

where y is an arbitrary fixed element of ℝ^{n}\{0}. Then from (5.3) and using (5.6) we obtain that ||S_{A}(t, τ)|| ≤ K||S_{A}(t_{i+j}^{+}, τ)|| = Kβ_{i+j} ≤ \frac{(K−1)^{j}}{K^{j−2}} β_{i}, hence

||S_{A}(t, τ)|| ≤ \frac{(K − 1)^{j}}{K^{j−2}} β_{i} for t ∈ [τ, t_{i+j+1}).

Since t_{k+1} −t_{k} < θ, k = 1, 2, ..., it follows that, for t ∈ [t_{i+j}, t_{i+j+1}), we have t − τ ≤ t_{i+j+1} − t_{i} < (j + 2)θ, that is, j > \frac{1}{θ}(t − τ) − 2. Therefore, we have that

||S_{A}(t, τ)|| ≤ \frac{K^{4}}{(K − 1)^{2}} β_{i}(1 − \frac{1}{K})^{\frac{1}{θ} (t−τ)} for t ∈ [t_{i+j}, t_{i+j+1}).

Further, we define the positive function λ(t), with −λ(t) ∈ R^{+}, as the solution of the inequality e_{−λ}(t, τ) ≥ (1 − \frac{1}{K})^{\frac{1}{θ} (t−τ)}, for t ∈ [t_{i+j}, t_{i+j+1}). Let

N = max\{\frac{K^{4}}{(K − 1)^{2}} β_{i}, sup_{τ≤t<t_{i}} \frac{||S_{A}(t, τ)||}{e_{−λ}(t, τ)}\}.

Then for all t, τ ∈ 𝕋_{+} with t ∈ 𝕋_{(τ)}, we obtain that

||S_{A}(t, τ)|| ≤ Ne_{−λ}(t, τ),

and so the theorem is proved. □

Remark 5.1. For example, when 𝕋 = ℝ, the solution of the inequality e_{−λ}(t, τ) = e^{−λ(t−τ)} ≥ (1 − \frac{1}{K})^{\frac{1}{θ} (t−τ)}, for τ ∈ [t_{i}, t_{i+1}), t ∈ [t_{i+j}, t_{i+j+1}] and j ≥ 0, is 0 ≤ λ ≤ −\frac{1}{θ} ln(1 − \frac{1}{K}), K > 1.

When

𝕋 = ℙ_{1,1} = ⋃_{k=0}^{∞}[2k, 2k + 1],

then e_{−λ}(t, τ) = (1−λ)^{j}e^{−λ(t−τ)}e^{λj} for τ ∈ [2i, 2i+1) and t ∈ [2(i+j), 2(i+j)+1]

with j ≥ 0. In this case, μ(t) = 0 if t ∈ ⋃_{k=0}^{∞}[2k, 2k + 1) and μ(t) = 1 if

t ∈ ⋃_{k=0}^{∞}{2k + 1}. It follows that −λ ∈ R^{+} if and only if λ ∈ [0, 1). Next, we

consider the function

f(λ) = (1 − λ)^{j}e^{−λ(t−τ)}e^{λj} − (1 − \frac{1}{K})^{\frac{1}{θ} (t−τ)}, λ ∈ [0, 1), K > 1.

Since f′(λ) = −[j + (1 − λ)(t − τ − j)](1 − λ)^{j−1}e^{−λ(t−τ)}e^{λj} it follows that f(λ) is decreasing on [0, 1). Then, f(0) = 1 − (1 − \frac{1}{K})^{\frac{1}{θ} (t−τ)} > 0 and lim_{λ↗1} f(λ) = −(1 − \frac{1}{K})^{\frac{1}{θ} (t−τ)} < 0, implies that there exists a unique λ_{0} ∈ (0, 1) such that f(λ_{0}) = 0. Therefore, the solution of the inequality

e_{−λ}(t, τ) = (1 − λ)^{j}e^{−λ(t−τ)}e^{λj} ≥ (1 − \frac{1}{K})^{\frac{1}{θ} (t−τ)}

for t ∈ [2(i + j), 2(i + j) + 1] with j ≥ 0 is 0 ≤ λ ≤ λ_{0}.

Corollary 5.1. Suppose that B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)), and there exists a positive constant θ such that t_{k+1} −t_{k} < θ, k = 1, 2, .... If the solution of initial value problem (5.1) is bounded for any c := {c_{k}}_{k=1}^{∞} ∈ l^{∞}(ℝ^{n}), then the dynamic system (3.2) is e.s.

Proof. From Theorem 5.1, exists a positive constants N = N(τ) ≥ 1, λ with −λ ∈ R^{+} such that ||S_{A}(t, τ)|| ≤ Ne_{−λ}(t, τ) for all t ∈ T_{(τ)}. For any τ ∈ 𝕋_{+}, the solution of (3.2) satisfies

||x(t)|| = ||S_{A}(t, τ)x(τ)|| ≤ ||S_{A}(t, τ)||·||x(τ)|| ≤ N||x(τ)||e_{−λ}(t, τ) for all t ∈ T_{(τ)}

and thus exponential stability is proved. □

Lemma 5.1. If there exist a constant θ > 0 such that t_{k+1} − t_{k} < θ, for k = 1, 2, ..., then for each constant λ > 0 with \frac{1}{λ} > θ we have that −λ ∈ R^{+}.

Proof. Since t_{k} and t_{k+1} are right-dense points then, for t ∈ [t_{k}, t_{k+1}], we have that t_{k} ≤ t ≤ σ(t) ≤ t_{k+1}. It follows that μ(t) = σ(t) − t ≤ t_{k+1} − t_{k} < θ for t ∈ [t_{k}, t_{k+1}]. Therefore, μ(t) < θ for t ∈ 𝕋_{+}. If \frac{1}{λ} > θ then we have that 1 − λμ(t) > 1 − \frac{1}{θ}μ(t) > 0 and thus −λ ∈ R^{+}. □

Theorem 5.2. Suppose that A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)), B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., and there exist positive constants θ, b, and M such that

t_{k+1} − t_{k} < θ, sup_{k≥1} ||B_{k}|| ≤ b, ∫_{t_{k}}^{t_{k+1}} ||A(s)||Δs ≤ M, k = 1, 2, ... (5.7)

If the solution of initial value problem (5.1) is bounded for any c := {c_{k}}_{k=1}^{∞} ∈ l^{∞}(ℝ^{n}), then there exists a positive constants N, λ with −λ ∈ R^{+} such that

||S_{A}(t, τ)|| ≤ Ne_{−λ}(t, τ) for all t ∈ 𝕋_{(τ)}.

Proof. Let k ∈ {1, 2, ...} be fixed. For an arbitrary fixed y ∈ ℝ^{n}\{0}, we define the sequence {c_{j}}_{j=1}^{∞} ∈ l^{∞}(ℝ^{n}) given by c_{k} = y and c_{j} = 0 if j ≠ k. Then x(t) = S_{A}(t, t_{k}^{+})y, t > t_{k}, is the solution of (5.1) with initial condition x(t_{k}^{+}) = 0.

From Theorem 5.1 we obtain that

||S_{A}(t, t_{k}^{+})|| ≤ N_{k}e_{−λ}(t, t_{k}^{+}) for all t > t_{k},

where

N_{k} = max\{\frac{K^{4}}{(K − 1)^{2}} ||S_{A}(t_{k+1}^{+}, t_{k}^{+})||, sup_{t_{k}≤t<t_{k+1}} \frac{||S_{A}(t, t_{k}^{+})||}{e_{−λ}(t, t_{k}^{+})}\}.

Now we have to show that N_{k} can be chosen independently of k.

From Corollary 3.2 and (5.7), we obtain that ||S_{A}(t, t_{k}^{+})|| ≤ e^{M}, t ∈ [t_{k}, t_{k+1}]. It follows that ||S_{A}(t_{k+1}^{+}, t_{k}^{+})|| ≤ ||I+B_{k}||·||S_{A}(t_{k+1}, t_{k}^{+})|| ≤ (b+1)e^{M}. On the other hand, since t − t_{k}^{+} ≤ t_{k+1} − t_{k} < θ, t ∈ [t_{k}, t_{k+1}] then, by Lemma 5.1, we can choose a constant λ with \frac{1}{λ} > θ such that that −λ ∈ R^{+}. Using the Bernoulli’s inequality, we have that

e_{−λ}(t, t_{k}^{+}) ≥ 1 − λ(t − t_{k}^{+}) ≥ 1 − λθ > 0,

and thus _{\frac{1}{e_{−λ}(t,t_{k}^{+})}} ≤ \frac{1}{1−λθ}. Hence ||S_{A}(t, t_{k}^{+})|| ≤ Ñe_{−λ}(t, τ), where Ñ =

max\{\frac{(b+1)K^{4}}{(K−1)^{2}} e^{M},\frac{b+1}{1−λθ} e^{M}\}.

Next, let τ ∈ 𝕋_{+} be arbitrary. Then there exists k ∈ {1, 2, ...} such that τ ∈ [t_{k}, t_{k+1}). Then we have

||S_{A}(t, τ)|| = ||S_{A}(t, t_{k}^{+})S_{A}(t_{k}^{+}, τ)|| ≤ ||S_{A}(t, t_{k}^{+})|| · ||S_{A}(t_{k}^{+}, τ)|| ≤
≤ Ñe_{−λ}(t, t_{k}^{+})Ñe_{−λ}S_{A}(t_{k}^{+}, τ) = Ñ^{2}e_{−λ}(t, τ).

Therefore ||S_{A}(t, τ)|| ≤ Ne_{−λ}(t, τ), with N = Ñ^{2}, and the theorem is proved. □

Corollary 5.2. Suppose that B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)), and there exist positive constants θ, b, and M such that

t_{k+1} − t_{k} < θ, sup_{k≥1} ||B_{k}|| ≤ b, ∫_{t_{k}}^{t_{k+1}} ||A(s)||Δs ≤ M, k = 1, 2, ...

If the solution of initial value problem (5.1) is bounded for any c := {c_{k}}_{k=1}^{∞} ∈ l^{∞}(ℝ^{n}), then the dynamic system (3.2) is u.e.s. □

Lemma 5.2. For any constant λ > 0 with −λ ∈ R^{+}, we have that

e_{−λ}(t, τ) ≤ e^{−λ(t−τ)}, for all τ, t ∈ 𝕋 with t ∈ 𝕋_{(τ)}.  (5.8)

Proof. Indeed, −λ ∈ R^{+} implies that 1 − λμ(t) > 0 for allt ∈ 𝕋_{+}. Since

lim_{u↘μ(s)} \frac{ln(1 − λu)}{u} = \{ \begin{matrix}−λ & if μ(s) = 0 \\ \frac{ln(1−λμ(s))}{μ(s)} ≤ −λ & if μ(s) ∈ (0, \frac{1}{λ}) \end{matrix}

then, using the explicit estimation of the modulus of the exponential function on time scales (see [24]), we have that

e_{−λ}(t, τ) = exp(∫_{τ}^{t} lim_{u↘μ(s)} \frac{ln(1 − λu)}{u}Δs)
≤ exp(− ∫_{τ}^{t} λΔs) ≤ e^{−λ(t−τ)},

and the inequality (5.8) is proved. □

Theorem 5.3. Suppose that B_{k} ∈ M_{n}(ℝ), k = 1, 2, ..., A ∈ C_{rd}R(𝕋_{+}, M_{n}(ℝ)), and there exist positive constants γ, θ, b, and M such that

γ < t_{k+1} − t_{k} < θ, sup_{k≥1} ||B_{k}|| ≤ b, ∫_{t_{k}}^{t_{k+1}} ||A(s)||Δs ≤ M, k = 1, 2, ...

If the solution of initial value problem (5.1) is bounded for any c := {c_{k}}_{k=1}^{∞} ∈ l^{∞}(ℝ^{n}), then the solution of (4.1) is bounded for each h ∈ BC_{rd}R(𝕋_{+}, ℝ^{n}).

Proof. Let i ∈ {1, 2, ...} be such that τ ∈ [t_{i}, t_{i+1}) By Theorem 5.2 there exist positive constants N and λ with −λ ∈ R^{+} such that

||S_{A}(t, τ)|| ≤ Ne_{−λ}(t, τ) for all t ∈ 𝕋_{(τ)}.

For every function h ∈ C_{rd}R(𝕋_{+}, ℝ^{n}), the corresponding solution x_{h} of (4.1) is given by (4.2). From (4.2) we obtain that

||x_{h}(t)|| ≤ ||S_{A}(t, τ)|| · ||η|| + ∫_{τ}^{t} ||S_{A}(t, σ(s))|| · ||h(s)||Δs

+ ∑_{τ<t_{j}<t} ||S_{A}(t, t_{j}^{+})|| · ||c_{j}|| ≤ N||η||e_{−λ}(t, τ)

+ ∫_{τ}^{t} N||h(s)||e_{−λ}(t, σ(s))Δs + ∑_{τ<t_{j}<t} e_{−λ}(t, t_{j}^{+})||c||_{l}^{∞}.

We have that

∫_{τ}^{t} N||h(s)||e_{−λ}(t, σ(s))Δs
≤ N||h|| ∫_{τ}^{t} e_{−λ}(t, σ(s))Δs = −\frac{N||h||}{λ}(e_{−λ}(t, τ) − e_{−λ}(t, t))
= \frac{N||h||}{λ}(1 − e_{−λ}(t, τ)) ≤ \frac{N||h||}{λ}.

Further, let t_{k} be the greatest of all t_{j} < t. Then

t − t_{k−1} ≥ t_{k} − t_{k−1} > γ, t − t_{k−2} > 2γ, t − t_{i} > (k − i)γ

and, by Lemma 5.2, we have that

∑_{τ<t_{j}<t} e_{−λ}(t, t_{j}^{+}) ≤ ∑_{j=i}^{∞} e^{−λγj} = \frac{1}{1 − e^{−λγ}} .

Therefore,

||x_{h}(t)|| ≤ N||η||e_{−λ}(t, τ) + \frac{N||h||}{λ} + \frac{N||c||_{l}^{∞}}{1 − e^{−λγ}} .

Since lim_{t→∞} e_{−λ}(t, τ) = 0 it follows that x_{h} is bounded. □

Example 5.1. We consider the linear impulsive dynamic system

\{ \begin{matrix}x^{Δ} = A(t)x, if t ∈ 𝕋_{(τ)}, t ≠ t_{2k} \\ x(t_{2k}^{+}) = (I + B_{2k})x(t_{2k}), k = 1, 2, ... \\ x(τ) = η, \end{matrix}  (5.9)

where A(t) = ( \begin{matrix}−α & 0 \\ 0 & −β \end{matrix} ), I +B_{2k} = ( \begin{matrix}a_{2k} & 0 \\ 0 & b_{2k} \end{matrix} ), β > α > 0 and t_{k} = k.

Also, we choose (a_{2k})_{k≥1} and (b_{2k})_{k≥1} such that a_{ij} := a_{2(i+1)}a_{2(i+2)}...a_{2(i+j)} ≤ a and b_{ij} := b_{2(i+1)}b_{2(i+2)}...b_{2(i+j)} ≤ b for each fixed i ≥ 0 and for j = 1, 2, .... If 𝕋 = ℝ, then then impulsive transition matrix associated with {B_{2k}, t_{2k}}_{k=1}^{∞} is given by

S_{A}(t, τ) = ( \begin{matrix}a_{ij}e^{−α(t−τ)} & 0 \\ 0 & b_{ij}e^{−β(t−τ)} \end{matrix} )

if τ ∈ [t_{2i}, t_{2i+2}) and t ∈ [t_{2(i+j)}, t_{2(i+j+1)}], where i ≥ 0 is fixed and j = 1, 2, ....

It follows that

∥S_{A}(t, τ)∥ ≤ Ne^{−α(t−τ)} for t ≥ τ,

where N = max{a, b}.

If 𝕋 = ℙ_{1,1}, then then impulsive transition matrix associated with {B_{2k}, t_{2k}}_{k=1}^{∞} is given by

S_{A}(t, τ) = ( \begin{matrix}a_{ij}(1 − α)^{j}e^{−α(t−τ)}e^{αj} & 0 \\ 0 & b_{ij}(1 − β)^{j}e^{−β(t−τ)}e^{βj} \end{matrix} )

= ( \begin{matrix}a_{ij}e_{−α}(t, τ) & 0 \\ 0 & b_{ij}e_{−β}(t, τ) \end{matrix} )

if τ ∈ [t_{2i}, t_{2i+1}) and t ∈ [t_{2(i+j)}, t_{2(i+j)+1}], where i ≥ 0 is fixed and j = 1, 2, ....

It follows that

∥S_{A}(t, τ)∥ ≤ Ne_{−α}(t, τ) for t ≥ τ,

where N = max{a, b}.

Next, the solution of the initial value problem

\{ \begin{matrix}x^{Δ} = A(t)x, t ∈ 𝕋_{(τ)}, t ≠ t_{2k} \\ x(t_{2k}^{+}) = x(t_{2k}) + B_{2k}x(t_{2k}) + c_{2k}, k = 1, 2, ... \\ x(τ^{+}) = 0, τ ≥ 0, \end{matrix}  (5.10)

is given by

x(t) = ∑_{l=1}^{j} S_{A}(t, t_{2(i+l)}^{+})c_{2(i+l)}

when τ ∈ [t_{2i}, t_{2i+2}) and t ∈ [t_{2(i+j)}, t_{2(i+j+1)}] if 𝕋 = ℝ or when τ ∈ [t_{2i}, t_{2i+1}) and t ∈ [t_{2(i+j)}, t_{2(i+j)+1}] if 𝕋 = ℙ_{1,1}, where i ≥ 0 is fixed and j = 1, 2, .... Then

∥x(t)∥ ≤ ∑_{l=1}^{∞} ∥S_{A}(t, t_{2(i+l)}^{+})∥ · ||c||_{l^{∞}}

≤ ∑_{l=1}^{∞} max{a_{il}, b_{il}}e^{−2α(l−1)}||c||_{l^{∞}},

where i ≥ 0 is fixed.

Therefore, if for each fixed i ≥ 0 we have that ∑_{l=1}^{∞} max{a_{il}, b_{il}}e^{−2α(l−1)} < ∞, then the solution of the initial value problem (5.10) is bounded for any

c := {c_{k}}_{k=1}^{∞} ∈ l^{∞}(ℝ^{n}). Moreover, that ∏_{l=1}^{∞} a_{il} < ∞, ∏_{l=1}^{∞} b_{il} < ∞ for each fixed

i ≥ 0, then

∥S_{A}(t, τ)∥ ≤ Ne_{−α}(t, τ) for t ≥ τ,

where N = max{a, b} and

e_{−α}(t, τ) = \{ \begin{matrix}e^{−α(t−τ)} & if t ∈ ℝ \\ (1 − α)^{l}e^{−α(t−τ)}e^{αl} & if t ∈ ℙ_{1,1} = ⋃_{l=0}^{∞}[2l, 2l + 1]. \end{matrix}

Consequently, the impulsive dynamic system (5.9) is uniformly exponentially stable.

6 Appendix on time scales analysis

We recall some basic definitions and results in the calculus on time scales analysis. We refer to [15, 16], and also to the paper [1, 2, 3, 8], for more information on analysis on time scales. A time scales 𝕋 is a nonempty closed subset of R, and the forward jump operator σ : 𝕋 → 𝕋 is defined by σ(t) := inf{s ∈ 𝕋; s > t} (supplemented by inf ∅ = sup 𝕋), the backward jump operator ρ : 𝕋 → 𝕋 is defined by ρ(t) := sup{s ∈ 𝕋; s < t} (supplemented by sup ∅ = inf 𝕋), while the graininess μ : 𝕋 →R_{+} is given by μ(t) := σ(t)−t. For our purpose, we will assume that the time scales 𝕋 is unbounded above, i.e., sup 𝕋 = ∞. The point t ∈ 𝕋 is left-dense, left-scattered, right-dense, right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t, σ(t) > t, respectively. A time scales 𝕋 is said to be discrete if t is left-scattered and rightscattered for each t ∈ 𝕋. The notations [a, b], [a, b), and so on, will denote time scales intervals such as [a, b] := {t ∈ 𝕋; a ≤ t ≤ b}, where a, b ∈ 𝕋. Let R^{n} be the space of n-dimensional column vectors x = col(x_{1}, x_{2}, ...x_{n}) with a norm || · ||. Also, by the same symbol || · || we will denote the corresponding matrix norm in the space M_{n}(R) of n × n matrices. We recall that ||A|| := sup{||Ax||; ||x|| ≤ 1} and the following inequality ||Ax|| ≤ ||A|| · ||x|| holds for all A ∈ M_{n}(R) and x ∈R^{n}.

Definition A.1. A function f : 𝕋 →R^{n} is said to be rd-contionous if (i) f is continuous at every right-dense point t ∈ 𝕋,

(ii) f(t^{−}) := lim_{s→t^{−}}f(s) exists and is finite at every left-dense point t ∈ 𝕋.

The set of all rd-continuous functions f : 𝕋 →R^{n} will be denoted by C_{rd}(𝕋,R^{n}). A function p : 𝕋 →R is said to be regressive (respectively positively regres-

sive) if 1 + μ(t)p(t) ≠ 0 (respectively 1 + μ(t)p(t) > 0) for all t ∈ 𝕋. The set R (respectively R^{+}) of all regressive (respectively positively regressive) functions from 𝕋 to R is an Abelian group with respect to the circle addition operation ⊕, given by

(p ⊕ q)(t) := p(t) + q(t) + μ(t)p(t)q(t).  (6.1)

The inverse element of p ∈ R is given by

(⊖p)(t) = −\frac{p(t)}{1 + μ(t)p(t)}  (6.2)

and so, the circle subtraction operation ⊖ is defined by

(p ⊖ q)(t) = (p ⊕ (⊖q))(t) = −\frac{p(t) − q(t)}{1 + μ(t)q(t)}.  (6.3)

The space of all rd-continuous and regressive functions from 𝕋 to R is denoted by C_{rd}R(𝕋,R). Also,

C_{rd}^{+}R(𝕋, ℝ) := {p ∈ C_{rd}R(𝕋, ℝ); 1 + μ(t)p(t) > 0 for all t ∈ 𝕋}.

The set of rd-continuous (respectively rd-continuous and regressive) functions A : 𝕋 → M_{n}(R) is denoted by C_{rd}(𝕋, M_{n}(R)) (respectively by C_{rd}R(𝕋, M_{n}(R))). We recall that a matrix-valued function A is said to be regressive if I + μ(t)A(t) is invertible for all t ∈ 𝕋, where I is the n×n identity matrix. Moreover, the set R(𝕋, M_{n}(R)) of all regressive matrix-valued functions is a group with respect to the addition operation ⊕ define

(A ⊕ B)(t) = A(t) + B(t) + μ(t)A(t)B(t)  (6.4)

for all t ∈ 𝕋. The inverse element of A ∈ R(𝕋, M_{n}(R)) is given by

(⊖A)(t) = −[I + μ(t)A(t)]^{−1}A(t) = −A(t)[I + μ(t)A(t)]^{−1}  (6.5)

for all t ∈ 𝕋.

Definition A.2. A function f : 𝕋 →R^{n} is said to be differentiable at t ∈ 𝕋, with delta-derivative f^{Δ}(t) ∈R^{n} if given ε > 0 there exists a neighborhood U of t such that, for all s ∈ 𝕋,

||f^{σ}(t) − f(s) − f^{Δ}(t)[σ(t) − s]|| ≤ ε|(t) − s|,

where f^{σ}(t) := f(σ(t)) for all t ∈ 𝕋.

We denote by C_{rd}^{1}(𝕋,R^{n}) the set of all functions f : 𝕋 →R^{n} that are differentiable on 𝕋 and its delta-derivative f^{Δ}(t) ∈ C_{rd}(𝕋,R^{n}).

Theorem A.1. ([1, 15]) Assume that f : 𝕋 →R^{n} and let t ∈ 𝕋.

(i) If f is differentiable at t, then f is continuous at t. (ii) If f is continuous at t and t is right-scattered, then f is differentiable at

t with

f^{Δ}(t) = \frac{f^{σ}(t) − f(t)}{σ(t) − t}.

(iii) If f is differentiable at t and t is right-dense, then

f^{Δ}(t) = lim_{s→t} \frac{f^{σ}(t) − f(s)}{t − s}.

(iv) If f is differentiable at t, then f^{σ}(t) = f(t) + μ(t)f^{Δ}(t).

(v) If f, g : 𝕋 →R^{n} are both differentiable at t, then the product f^{T} g is also

differentiable at t and (f^{T} g)^{Δ}(t) = f^{T} (t)g^{Δ}(t) + (f^{T})^{Δ}(t)g^{σ}(t).

Theorem A.2. ([15]) If A, B : 𝕋 → M_{n}(R) are differentiable, then

(i) A^{σ}(t) = A(t) + μ(t)A^{Δ}(t) for all t ∈ 𝕋; (ii) (A^{T})^{Δ} = (A^{Δ})^{T}; (iii) (A + B)^{Δ} = A^{Δ} + B^{Δ}, and (AB)^{Δ} = A^{Δ}B^{σ} + AB^{Δ} = A^{σ}B^{Δ} + A^{Δ}B; (iv) (A^{−1})^{Δ} = −(A^{σ})^{−1}A^{Δ}A^{−1} = −A^{−1}A^{Δ}(A^{σ})^{−1}if AA^{σ} is invertible;

(v) (AB^{−1})^{Δ} = (A^{Δ} − AB^{−1}B^{Δ})(B^{σ})^{−1} = [A^{Δ} − (AB^{−1})^{σ}B^{Δ}]B^{−1} if BB^{σ} is invertible.

Definition A.3. Let f ∈ C_{rd}(𝕋,R^{n}). A function g : 𝕋 → ℝ^{n} is called the antiderivative of f on 𝕋 if it is differentiable on 𝕋 and satisfies g^{Δ}(t) = f(t) for all t ∈ 𝕋. In this cases, we define

∫_{a}^{t} f(s)Δs = g(t) − g(a), a, t ∈ 𝕋.

Theorem 3. (Existence of Antiderivatives, ) Every function f ∈ C_{rd}(T,R^{n}) has an antiderivative. In particular if τ ∈ T, then the function F : T → R^{n} defined by

F (t) := ∫_{τ}^{t} f(s)Δs for t ∈ 𝕋

is an antiderivative of f.

Theorem A.4. ([2, 15]) If a, b, c ∈ 𝕋, α, β ∈ ℝ, and f, g ∈ C_{rd}(𝕋, ℝ^{n}), then

EJQTDE, 2010 No. 11, p. 25 (i) ∫_{a}^{b}(αf + βg)(t)Δt = α ∫_{a}^{b} f(t)Δt + β ∫_{a}^{b} g(t)Δt; (ii) ∫_{a}^{b} f(t)Δt = − ∫_{b}^{a} f(t)Δt; (iii) ∫_{a}^{σ(a)} f(t)Δt = μ(a)f(a); (iv) ∫_{a}^{b} f(t)Δt = ∫_{a}^{c} f(t)Δt + ∫_{c}^{b} f(t)Δt;

(v) ∫_{a}^{b} f^{T} (t)g^{Δ}(t)Δt = (f^{T} g)(b) − (f^{T} g)(a) − ∫_{a}^{b}(f^{T})^{Δ}(t)g(σ(t))Δt; (vi) ∥∫_{a}^{b} f(t)Δt∥ ≤ ∫_{a}^{b} ||f(t)||Δt;

Let p ∈ C_{rd}R(𝕋,R) and μ(t) ≠ 0 for all t ∈ 𝕋. Then the exponential function on 𝕋 is defined by

e_{p}(t, s) = exp (∫_{s}^{t} ξ_{μ(t)}(p(τ))Δτ) with ξ_{h}(z) := \{ \begin{matrix}\frac{ln(1+hz)}{h} & if h ≠ 0 \\ z & if h = 0, \end{matrix}

and it is the unique solution of the initial value problem y^{Δ} = p(t)y, y(s) = 1.

Theorem A.5. ([15]) If p, q ∈ C_{rd}R(𝕋,R) then the following hold:

(i) e_{0}(t, s) = 1 and e_{p}(t, t) = 1; (ii) e_{p}(σ(t), s) = [1 + μ(t)p(t)]e_{p}(t, s); (iii) \frac{1}{e_{p}(t,s)} = e_{p}(s, t) = e_{⊖p}(t, s); (iv) e_{p}(t, s)e_{p}(s, r) = e_{p}(t, r);

(v) e_{p}(t, s)e_{q}(t, s) = e_{p⊕q}(t, s) and \frac{e_{p}(t,s)}{e_{q}(t,s)} = e_{p⊖q}(t, s); (vi) (\frac{1}{e_{p}(·,s)})^{Δ} = −\frac{p(t)}{e_{p}^{σ}(·,s)}; (vii) If p ∈ R^{+} then e_{p}(t, s) > 0 for all t, s ∈ 𝕋; (viii) ∫_{a}^{b} p(s)e_{p}(c, σ(s))Δs = e_{p}(c, a) − e_{p}(c, b).

Lemma A.2. ([15, Theorem 6.1]) Let τ ∈ 𝕋, y, b ∈ C_{rd}R(𝕋,R) and p ∈ C_{rd}^{+}R(𝕋,R). Then

y^{Δ}(t) ≤ p(t)y(t) + b(t) for all t ∈ 𝕋

implies

y(t) ≤ y(τ)e_{p}(t, τ) + ∫_{τ}^{t} e_{p}(t, σ(s))b(s)Δs for all t ∈ 𝕋.

Lemma A.3. (Bernoulli’s inequality, [15, Theorem 6.2]) Let α ∈ R with α ∈ R^{+}. Then

e_{α}(t, τ) ≥ 1 + α(t − τ), for all t ∈ 𝕋_{(τ)}.

Lemma A.4. (Gronwall’s inequality, [15, Theorem 6.4]) Let τ ∈ 𝕋, y, b ∈ C_{rd}R(𝕋,R) and p ∈ C_{rd}^{+}R(𝕋,R), p ≥ 0. Then

y(t) ≤ b(t) + ∫_{τ}^{t} y(s)p(s)Δs for all t ∈ 𝕋

implies

y(t) ≤ b(t) + ∫_{τ}^{t} e_{p}(t, σ(s))b(s)Δs for all t ∈ 𝕋.

Lemma A.5. ([31]) Let τ ∈ 𝕋_{+}, y ∈ C_{rd}R(𝕋_{+},R), p ∈ C_{rd}^{+}R(𝕋_{+},R) and c_{k}, d_{k} ∈R_{+}, k = 1, 2, .... Then

\{ \begin{matrix}y^{Δ}(t) ≤ p(t)y(t) + b(t) , t ∈ 𝕋_{(τ)}, t ≠ t_{k} \\ y(t_{k}^{+}) ≤ c_{k}y(t_{k}) + d_{k}, k = 1, 2, ... \end{matrix}

implies

y(t) ≤ y(τ) ∏_{τ<t_{k}<t} c_{k}e_{p}(t, τ) + ∑_{τ<t_{k}<t}( ∏_{t_{k}<t_{j}<t} c_{j}e_{p}(t, t_{k}))d_{k}

+ ∫_{τ}^{t} ∏_{s<t_{k}<t} c_{k}e_{p}(t, σ(s))b(s)Δs.

Acknowledgement

The first author’s research was supported by the Government College University, Abdus Salam School of Mathematical Sciences, (ASSMS), Lahore, Pakistan.

References

[1] R. P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications, Results Math. 35 (1–2) (1999) 3–22.

[2] R. P. Agarwal, M. Bohner, Quadratic functionals for second-order matrix equations on time scales, Nonlinear Analysis 33(1998) 675-692.

[3] R. P. Agarwal, M. Bohner, D. O’Regan, A. Peterson, Dynamic equations on time scales: A survey, J. Comput. Appl. Math. 141 (2002) 1–26.

[4] M. U. Akhmet, On the general problem of stability for impulsive differential equations, J. Math. Anal. Appl. 288 (2003) 182–196.

[5] M. U. Akhmet, J. Alzabut, A. Zafer, Perron’s theorem for linear impulsive differential equations with distributed delay, J. Comput. Appl. Math. 193(2006)204-218.

[6] A. Anokhin, L. Berezansky, E. Braverman, Exponential stability of linear delay impulsive differential equations, J. Math. Anal. Appl. 193(1995) 923-991.

[7] F. M. Atici, D. C. Biles, First and second order dynamic equations with impulse, Adv. Difference Equ. 2 (2005) 119–132.

[8] A. Aulbzch and S. Hilger, A unified approach to continuous and discrete dynamics, J. Qualitative Theory of Differential Equations 53, Szeged, Hungary, 1988.

[9] A. Aulbach and S. Hilger, Nonlinear Dynamics and Quantum Dynamical Systems, Akademic-Verlag, Berlin, 1990.

[10] D. D. Bainov and S. I. Kostadinov, Abstract Impulsive Differential Equations, World Scientific, New Jersey, 1995.

[11] D. D. Bainov, P. S. Simeonov, Impulsive differential equations: periodic solutions and applications, New York, Longman Scientific and Technical Group. Limited, 1993.

[12] M. Benchohra, J. Henderson and S. K. Ntouyas, Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications 2, Hindawi Publ. Corp., New York, 2006.

[13] M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahab, On first order impulsive dynamic equations on time scales, J. Differ. Equ. Appl. 10 (2004) 541–548.

[14] L. Berezansky, E. Braverman, Exponential boundedness of solutions for impulsive delay differential equations, Appl. Math. Lett. 9(6)(1996) 91-95.

[15] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäuser, Boston, MA, 2001.

[16] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.

[17] M. Bohner, A. Peterson, First and second order linear dynamic equations on time scales, J. Differ. Equ. Appl. 7 (2001) 767–792.

[18] M. Bohner and G. S. Guseinov, Improper integrals on time scales, in: Dynamic Equations on Time Scales, Dynam. Systems Appl. 12 (2003) 1-2, 45–65 (special issue).

[19] C. Buşe, On nonuniform asymptotic stability of impulse differential equations, Demonstratio Mathematica, 31(3)(1998) 485-497.

[20] S. K. Choi, N. J. Koo, D. M. Im, h-Stability of linear dynamic equations, J. Math. Anal. Appl. 324 (2006) 707–720.

[21] J. J. DaCunha, Stability for time varying linear dynamic systems on time scales, J. Comput. Appl. Math. 176 (2005) 381–410.

[22] Ju. L. Daleckiĭ, M. G. Kreĭn, Stability of Solution of Differential Equations in Banach Spaces, Volume 43, Transl. Math. Monographs, Amer. Math. Soc., Providence, Rhode Island, 1974.

[23] N. H. Du, L. H. Tien, On the exponential stability of dynamic equations on time scales, J. Math. Anal. Appl. 331 (2007) 1159–1174.

[24] T. Gard, J. Hoffacker, Asymptotic behavior of natural growth on time scales, Dynam. Systems Appl. 12 (2003) 131–147.

[25] F. Genga, Y. Xu, D. Zhu, Periodic boundary value problems for first-order impulsive dynamic equations on time scales, Nonlinear Analysis 69 (2008) 4074–4087.

[26] G. S. Guseinov, Integration on time scales, J. Math. Anal. Appl. 285 (2003) 107–127.

[27] A. Halanay, D. Wexler, Qualitative theory of systems with impulses, Editura Acad. Rep. Soc. Romania, Bucuresti, 1968 (in Romanian).

[28] S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, PhD thesis, Univ. Würzburg, 1988.

[29] S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18–56.

[30] J. Hoffacker, C.C. Tisdell, Stability and instability for dynamic equations on time scales, Comput. Math. Appl. 49 (2005) 1327–1334.

[31] M. Huang, W. Feng, Oscillation criteria for impulsive dynamic equations on time scales, Electron. J. Diff. Equ. 2007(2007), no.72, 1-13.

[32] V. Lakshmikantham, S. Sivasundaram and B. Kaymakalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996.

[33] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.

[34] V. Lakshmikantham, A. S. Vatsala, Hybrid systems on time scales, J. Comput. Appl. Math. 141 (2002) 227–235.

[35] H. Liu, X. Xiang, A class of the first order impulsive dynamic equations on time scales, Nonlinear Analysis 69 (2008) 2803–2811.

[36] Y. Ma, J. Sun, Stability criteria for impulsive systems on time scales, J. Comput. Appl. Math. 213 (2008) 400 – 407.

[37] J. J. Nieto, Impulsive resonance periodic problems of first order, Appl. Math.Lett.15(4)(2002)489-493.

[38] A. Peterson, R. F. Raffoul, Exponential stability of dynamic equations on time scales, Adv. Difference Equ. Appl. (2005) 133–144.

[39] M. Pituk, A criterion for the exponential stability of linear difference equations, Appl. Math.Lett. 17(2004) 779-783.

[40] C. Pötzsche, Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients, J. Math. Anal. Appl. 289 (2004) 317–335.

[41] C. Pötzsche, S. Siegmund, F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales, Discrete Contin. Dyn. Syst. 9 (2003) 1223–1241.

[42] A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.

(Received August 30, 2009)