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Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

  • Zhien Li and Chao Wang EMAIL logo
Published/Copyright: May 26, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

Keywords: quaternion dynamic equations; Cauchy matrix; Liouville formula; impulsive; time scales
MSC 2010: 34A37; 34N05; 11R52

1 Introduction

In 1843, Hamilton initiated the concept of quaternions that extends the complex numbers to four-dimensional space [1]. Quaternions are 4-vectors, whose multiplication is determined by a noncommutative division algebra. Let the quaternions q = ( q 0 , q 1 , q 2 , q 3 ) ∈ ℝ 4 be

q = q 0 + q 1 i + q 2 j + q 3 k ,

where q 0 , q 1 , q 2 , q 3 ∈ ℝ and i, j, and k satisfy the following multiplication:

i 2 = j 2 = k 2 = − 1 , j k = − k j = i , k i = − i k = j , i j = − j i = k .

In the real-world applications, quaternions are superior to the real-valued vectors in description of the phenomena in physics and life sciences [2]. In fact, there exists the quaternionic differential equation structure in many research fields such as differential geometry, fluid mechanics, attitude dynamics, and quantum mechanics, and many interesting phenomena under the quaternionic background have attracted many researchers [3,4,5,6,7]. To the best of our knowledge, there are few research results on the theory of quaternion dynamic equations on time scales [8].

To study the dynamic equations on hybrid domains, in 1988, Stefan Hilger introduced the theory of time scales, which provides an effective way to unify various hybrid domain analysis and has recently received a lot of attention [9,10]. The non-quaternion dynamic equations and applications on time scales in various fields were well studied, and many results were obtained [11,12,13,14]. It is well known that a time scale T is an arbitrary nonempty closed subset of the reals, by choosing the time scale to be the set of real numbers, the general results yields the results concerning the different types of dynamic equations, for example, let T = ℝ , then the dynamic equations will turn into differential equations, if T = h Z , then the same results yield a result for difference equations with h-step. However, since there are many other time scales compared with real numbers and integers such as the quantum time scale T = q Z , the hybrid domains T = { h Z } ∪ { q Z ¯ } , etc., one can obtain a much more general result by using the theory of time scales.

On the other hand, the impulsive dynamic equations play a vital role in describing the natural phenomena with sudden changes; it is a hot topic of the research of impulsive dynamic systems since the instantaneous change caused by impulses has a very significant meaning in explaining and mastering the change rule of the object. Due to this reason, there have been many literature in this field [15,16,17,18].

Nevertheless, there are no research results related to the Cauchy matrix and Liouville formula for the theory of quaternion impulsive dynamic equations on time scales, which will lead to many difficulties in studying the quaternion impulsive dynamic equations on complex hybrid domains. To fill this gap, in Sections 3 and 4, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are derived; based on them, the existence, uniqueness, and expressions of their solutions are also obtained for their scalar form and matrix form, respectively. In each section, several concrete examples and counter examples are provided to analyze the feasibility of our obtained results.

2 Preliminaries

We denote the space of quaternion by ℍ . For any q = x 0 + x 1 i + x 2 j + x 3 k ∈ ℍ , the conjugate, the real part, and the imaginary part of q are, respectively, as follows:

q ¯ = x 0 − x 1 i − x 2 j − x 3 k , ℜ ( q ) = x 0 , ℑ ( q ) = x 1 i + x 2 j + x 3 k .

The quaternion multiplication is a simple noncommutative division algebra, but the real and quaternion is commutable, i.e., if t ∈ ℝ and q ∈ ℍ , then tq = qt. Besides,

q ¯ q = x 0 2 + x 1 2 + x 2 2 + x 3 2 = | q | 2 , q h ¯ = h q ¯ , q − 1 = q ¯ | q | 2 , ℜ ( q p ) = ℜ ( p q ) .

Similar to Definition 5.18 from [9], we can also introduce the following definition of quaternion-valued matrix exponential function.

Definition 2.1

Let f : T → ℍ , we define the exponential function e f (t,t 0) by the solution of the initial value problem:

x Δ ( t ) = f ( t ) x ( t ) , x ( t 0 ) = 1 .

Also, let A : T → ℍ n × n , the matrix exponential function e A (t,t 0) is defined by the solution of the initial value problem:

X Δ ( t ) = A ( t ) X ( t ) , X ( t 0 ) = I ,

where I is the n × n identity matrix.

Definition 2.2

For A : T → ℍ n × n , where A(t) = [a rs (t)] n×n , 1 ≤ r, s ≤ n, if every a rs (t) is rd-continuous, then A is said to be rd-continuous quaternion-valued matrix, the collection of all such matrix functions is denoted by ℛ r d .

Definition 2.3

[19] For every quaternion function matrix A ∈ ℛ r d , it can be expressed uniquely in the form of:

A ( t ) = ℭ 1 ( A ( t ) ) + ℭ 2 ( A ( t ) ) j , where ℭ 1 ( A ( t ) ) , ℭ 2 ( A ( t ) ) ∈ ℂ n × n .

Hence, we can define G : ℍ n × n → ℂ n × n by

G ( A ( t ) ) = [ ℭ 1 ( A ( t ) ) ℭ 2 ( A ( t ) ) − ℭ 2 ( A ( t ) ) ¯ ℭ 1 ( A ( t ) ) ¯ ] ,

where G ( A ( t ) ) is called the complex adjoint function matrix of the quaternion function matrix A(t).

Denote ℌ ( A ( t ) ) by

ℌ ( A ( t ) ) = [ ℭ 1 ( A ( t ) ) ℭ 1 ( A ( t ) ) ¯ + ℭ 2 ( A ( t ) ) ℭ 2 ( A ( t ) ) ¯ ] .

Remark 2.1

For A ∈ ℛ r d , let A(t) = A 0(t) + A 1(t)i + A 2(t)j + A 3(t)k, where A n 0 : T → ℝ n × n , n 0 = 0, 1, 2, 3, by Definition 2.3 the following hold:

ℭ 1 ( A ( t ) ) ℭ 1 ( A ( t ) ) ¯ = A 0 ( t ) A 0 ( t ) + A 1 ( t ) A 1 ( t ) + [ A 0 ( t ) A 1 ( t ) − A 1 ( t ) A 0 ( t ) ] i , ℭ 2 ( A ( t ) ) ℭ 2 ( A ( t ) ) ¯ = A 2 ( t ) A 2 ( t ) + A 3 ( t ) A 3 ( t ) + [ A 2 ( t ) A 3 ( t ) − A 3 ( t ) A 2 ( t ) ] i .

Hence, ℌ ( A ( ⋅ ) ) ∈ ℂ n × n . Moreover, ℌ ( A ( ⋅ ) ) ∈ ℝ n × n if and only if

(2.1) A 0 ( t ) A 1 ( t ) − A 1 ( t ) A 0 ( t ) + A 2 ( t ) A 3 ( t ) − A 3 ( t ) A 2 ( t ) = 0 .

Example 2.1

For some t 0 ∈ T , let

A ( t 0 ) = [ 1 + i 2 + i 1 3 + i ] + [ 1 + i 3 + i 1 i ] j .

Hence,

ℭ 1 ( A ( t 0 ) ) ℭ 1 ( A ( t 0 ) ) ¯ = [ 1 + i 2 + i 1 3 + i ] [ 1 − i 2 − i 1 3 − i ] = [ 4 + i 10 + 2 i 4 12 − i ] , ℭ 2 ( A ( t 0 ) ) ℭ 2 ( A ( t 0 ) ) ¯ = [ 1 + i 3 + i 1 i ] [ 1 − i 3 − i 1 − i ] = [ 5 + i 5 − i 1 4 − i ] .

Therefore,

ℌ ( A ( t 0 ) ) = [ 9 + 2 i 15 + i 5 16 − 2 i ] ∈ ℂ 2 × 2 .

Definition 2.4

[8] For any A ∈ ℛ r d and ℌ ( A ( ⋅ ) ) ∈ ℝ n × n , we define the determinant of the quaternion function matrix by:

ddet A ( t ) := det [ ℌ ( A ( t ) ) ] .

Remark 2.2

According to Example 2.1, ℌ ( A ( ⋅ ) ) cannot be always a real-valued matrix. Hence, the condition ℌ ( A ( ⋅ ) ) ∈ ℝ n × n in Definition 2.4 is necessarily required.

Remark 2.3

If A : T → ℝ n × n , then ddet A(t) = det A(t) det A(t).

We introduce the notation D = { A ∈ ℛ r d : ℌ ( A ( ⋅ ) ) ∈ ℝ n × n } .

For any A ( ⋅ ) = [ a r h ( ⋅ ) ] n × n ∈ ℍ n × n and a r h , x : T → ℍ , let

a r h ( t ) = a r h 0 ( t ) + a r h 1 ( t ) i + a r h 2 ( t ) j + a r h 3 ( t ) k , x ( t ) = x 0 ( t ) + x 1 ( t ) i + x 2 ( t ) j + x 3 ( t ) k ,

where a rhs , x s : T → ℝ and 1 ≤ r, h ≤ n, s = 0, 1, 2, 3. Define

∥; A ( t ) ∥; = ∑ r , h =1 n ∑ s = 0 3 | a r h s ( t ) | , ∥; x ( t ) ∥; = ∑ s = 0 3 | x s ( t ) | .

Definition 2.5

[8] Let f(t) = f 0(t) + f 1(t)i + f 2(t)j + f 3(t)k and f r : T → ℝ be rd-continuous for each r = 0, 1, 2, 3, the integral of the function f(t) is defined as follows:

∫ t 0 t f ( τ ) Δ τ = ∫ t 0 t f 0 ( τ ) Δ τ + i ∫ t 0 t f 1 ( τ ) Δ τ + j ∫ t 0 t f 2 ( τ ) Δ τ + k ∫ t 0 t f 3 ( τ ) Δ τ ,

where i, j, and k are the quaternion imaginary units.

Definition 2.6

[19] Let A ( ⋅ ) = [ a r h ( ⋅ ) ] n × n ∈ ℛ r d and a r h : T → ℍ , where 1 ≤ r, h ≤ n, the integral of the matrix function A(t) is defined as follows:

∫ t 0 t A ( τ ) Δ τ = [ ∫ t 0 t a s r ( τ ) Δ τ ] .

3 Quaternion scalar impulsive dynamic equation

Now, we consider the impulsive dynamic equations on a time scale as follows:

(3.1) { x Δ ( t ) = f ( t ) x ( t ) + h ( t ) , t ≠ t n , Δ ˜ x ( t ) = m n x ( t ) , t = t n ,

where f : T → ℍ , m n ∈ ℍ , t n ∈ T , Δ ˜ x ( t ) = x ( σ ( t + ) ) − x ( t ) .

Remark 3.1

In (3.1), if t n is the right-dense point, then Δ ˜ x ( t ) = x ( t + ) − x ( t ) ; if t n is the right-scattered point, then Δ ˜ x ( t ) = x ( σ ( t ) ) − x ( t ) .

Lemma 3.1

Let F ( t ) = ∫ t 0 t f ( τ ) ∫ t 0 τ g ( η ) Δ η Δ τ and t 0 ∈ T be fixed, where f , g : T → ℍ , then F : T → ℍ is differentiable at t with

F Δ ( t ) = f ( t ) ∫ t 0 t g ( η ) Δ η .

Moreover, if R ( t ) = ∫ t 0 t f ( η n ) ∫ t 0 η n f ( η n − 1 ) … ∫ t 0 η 2 f ( η 1 ) Δ η 1 … Δ η n − 1 Δ η n , then

(3.2) R Δ ( t ) = f ( t ) ∫ t 0 t f ( η n − 1 ) … ∫ t 0 η 2 f ( η 1 ) Δ η 1 … Δ η n − 1 .

Proof

For the right-scattered point t ∈ T , the Δ-derivative of F(t) can be calculated as follows:

F Δ ( t ) = ∫ t 0 σ ( t ) f ( τ ) ∫ t 0 τ g ( η ) Δ η Δ τ − ∫ t 0 t f ( τ ) ∫ t 0 τ g ( η ) Δ η Δ τ μ ( t ) = ∫ t σ ( t ) f ( τ ) Δ τ ∫ t 0 τ g ( η ) Δ η μ ( t ) = [ ∫ t σ ( t ) f 0 ( τ ) Δ τ + i ∫ t σ ( t ) f 1 ( τ ) Δ τ + j ∫ t σ ( t ) f 2 ( τ ) Δ τ + k ∫ t σ ( t ) f 3 ( τ ) Δ τ ] ∫ t 0 τ g ( η ) Δ η μ ( t ) = [ f 0 ( t ) + i f 1 ( t ) + j f 2 ( t ) + k f 3 ( t ) ] ∫ t 0 t g ( η ) Δ η = f ( t ) ∫ t 0 t g ( η ) Δ η .

For the right-dense point t ∈ T , the derivative of F(t) is obvious. Similar to the above calculation, by taking F(t) = R(t), we can obtain (3.2). The proof is complete.□

Now, consider the following homogeneous linear dynamic equations:

(3.3) { x Δ ( t ) = f ( t ) x ( t ) , x ( t 0 ⁎ ) = x 0 ,

where f : T → ℍ is rd-continuous, x 0 ∈ ℍ , t 0 ⁎ ∈ T .

Lemma 3.2

For (3.3), if f is uniformly bounded on T , i.e., there exists some constant M f > 0, such that ∥; f ( t ) ∥; ≤ M f for all t ∈ T , then the solution x(t) of the initial value problem of (3.3) is rd-continuous and given by

x ( t ) = ( 1 + ∑ n = 1 ∞ c n ( t ) ) x 0 ,

where

c n ( t ) = ∫ t 0 ⁎ t f ( t n ) ∫ t 0 ⁎ t n f ( t n − 1 ) … ∫ t 0 ⁎ t 2 f ( t 1 ) Δ t 1 … Δ t n − 1 Δ t n .

Proof

Let h be constant with h > 0. For t 0 ⁎ ≤ t < t 0 ⁎ + h , we have

∥; c n ∥; ≤ ∫ t 0 ⁎ t M f ∫ t 0 ⁎ t n M f … ∫ t 0 ⁎ t 2 M f Δ t 1 … Δ t n − 1 Δ t n = M f n h n ( t , t 0 ⁎ ) ≤ M f n ( t − t 0 ⁎ ) n n ! = M f n h n n ! .

By the Weierstrass theorem, the series ∑ n = 1 ∞ M f n h n n ! is convergent (say it is convergent to a ∈ ℝ + ), which implies that the series { ∑ n = 1 ∞ c n } is uniformly convergent on T .

Next, we show that the function x(t) is rd-continuous. For the right-dense t r ∈ T , ∀ ε > 0 , there exists δ ( ε ) = ε / a M f > 0 , such that | t − t r | < δ ( ε ) , we have

∥; x ( t ) − x ( t r ) ∥; = ∥; ∑ n =1 ∞ ∫ t r t f ( t n ) ∫ t 0 ⁎ t n f ( t n − 1 ) … ∫ t 0 ⁎ t 2 f ( t 1 ) Δ t 1 … Δ t n − 1 Δ t n x 0 ∥; ≤ ∑ n =1 ∞ M f n | t − t r | ( t − t 0 ⁎ ) n − 1 ( n − 1)! ≤ δ ( ε ) ∑ n =1 ∞ M f n ( t − t 0 ⁎ ) n − 1 ( n − 1)! ≤ ε a M f M f ∑ n =1 ∞ M f n − 1 h n − 1 ( n − 1)! < ε .

Thus, x(t r ) is continuous at right-dense. Moreover, since the function f(t) is rd-continuous, it follows that x(t) has the finite left-side limit at a left-dense point. Therefore, x(t) is rd-continuous.

On the other hand, by Lemma 3.1, we can get c n Δ ( t ) = f ( t ) c n − 1 ( t ) , and hence, x ( t ) = x 0 + ∑ n = 1 ∞ c n ( t ) x 0 is Δ-differentiable with

x Δ ( t ) = ∑ n = 1 ∞ f ( t ) c n − 1 ( t ) x 0 ,

where c 0 = 1, hence x Δ ( t ) = f ( t ) x ( t ) . Therefore, the function series x ( t ) = x 0 + ∑ n = 1 ∞ c n x 0 is a solution of (3.3), according to the continuation theorem of solutions for dynamic equations, x(t) is a solution for (3.3) on T . Furthermore, we assume that x 1 and x 2 are two solutions of (3.3), then

∥; x 1 ( t ) − x 2 ( t ) ∥; ≤ ∫ t 0 t ∥; f ( τ ) ∥; ∥; x 1 ( τ ) − x 2 ( τ ) ∥; Δ τ ≤ M ∫ t 0 t ∥; x 1 ( τ ) − x 2 ( τ ) ∥; Δ τ .

By Corollary 6.7 from [9] (Bellman inequality on time scale), we can get ∥; x 1 ( t ) − x 2 ( t ) ∥; = 0 . Therefore, the solution of (3.3) is unique. The proof is complete.□

Theorem 3.1

Let f : T → ℍ be rd-continuous on T and (3.3) with the initial condition x ( t 0 ⁎ ) = 1 has the solution with the exponential form as follows:

e f ( t , t 0 ⁎ ) = 1 + ∑ n = 1 ∞ ∫ t 0 ⁎ t f ( τ n ) ∫ t 0 ⁎ τ n f ( τ n − 1 ) … ∫ t 0 ⁎ τ 2 f ( τ 1 ) Δ τ 1 … Δ τ n − 1 Δ τ n .

Proof

Let

C n ( t ) = ∑ n =1 ∞ ∫ t 0 ⁎ t f ( τ n ) ∫ t 0 ⁎ τ n f ( τ n − 1 ) … ∫ t 0 ⁎ τ 2 f ( τ 1 ) Δ τ 1 … Δ τ n − 1 Δ τ n

for n ≥ 1, by Lemma 3.1, we can get

C n Δ ( t ) = f ( t ) C n − 1 ( t ) .

By Lemma 3.1, we can get 1 + ∑ n = 1 ∞ C n ( t ) is a unique solution of (3.3) with x ( t 0 ⁎ ) = 1 . Hence, by Definition 2.1, we obtain

e f ( t , t 0 ⁎ ) = 1 + ∑ n =1 ∞ ∫ t 0 ⁎ t f ( τ n ) ∫ t 0 ⁎ τ n f ( τ n − 1 ) … ∫ t 0 ⁎ τ 2 f ( τ 1 ) Δ τ 1 … Δ τ n − 1 Δ τ n .

The proof is complete.□

Remark 3.2

From Theorem 3.1, for any f : T → ℍ and s , t ∈ T , we can easily obtain (i) ef(t,t) = 1; (ii) e f  (σ(t),s) − e f  (t,s) = μ(t)f(t)e f  (t,s); and (iii) e f  (t,s) ≠ e f  (s,t).

Now, consider the homogeneous linear impulsive dynamic equations as follows:

(3.4) { x Δ ( t ) = f ( t ) x ( t ) , t ≠ t n , Δ ˜ x ( t ) = m n x ( t ) , t = t n , n ∈ Z , x ( t 0 ⁎ ) = x 0 ,

where t n ∈ { t 1 , t 2 , … , t n 0 } ⊂ ( t 0 ⁎ , t ) T , for t > t 0 ⁎ , f : T → ℍ , m n ∈ ℍ , Δ ˜ x ( t ) = x ( σ ( t + ) ) − x ( t ) .

Lemma 3.3

The solution of (3.4) can be given as

x ( t ) = { [ 1 + ∑ r = 1 ∞ c 1, p ] x 0 , t 0 ⁎ ≤ t < σ ( t 1 + ) , [ 1 + ∑ r = 1 ∞ c s , p ] ∏ v = s − 1 1 (1 + m v ) [ 1 + ∑ l = 1 ∞ C v , l ] x 0 , σ ( t s − 1 + ) ≤ t < σ ( t s + ) , 1 < s ≤ n 0 , [ 1 + ∑ r = 1 ∞ c s , p ] ∏ v = s 1 (1 + m v ) [ 1 + ∑ l = 1 ∞ C v , l ] x 0 , t = σ ( t s + ) , 1 < s ≤ n 0 , [ 1 + ∑ r =1 ∞ c n 0 + 1, p ] ∏ v = n 0 1 (1 + m v ) [ 1 + ∑ l = 1 ∞ C v , l ] x 0 , σ ( t n 0 + ) < t ,

where c s , r = ∫ σ ( t s − 1 + ) t f ( τ r ) ∫ σ ( t s − 1 + ) τ r f ( τ r − 1 ) … ∫ σ ( t s − 1 + ) τ 2 f ( τ 1 ) Δ τ 1 … Δ τ r − 1 Δ τ r and C s,r = C s,r (t s ), 1 < s ≤ n 0.

Proof

By Lemma 3.2, for t 0 ⁎ ≤ t < σ ( t 1 + ) , we have

x ( t ) = x 0 + ∑ r = 1 ∞ c 1 r x 0 , c 1 , r = ∫ t 0 ⁎ t f ( τ r ) ∫ t 0 ⁎ τ r f ( τ r − 1 ) … ∫ t 0 ⁎ τ 2 f ( τ 1 ) Δ τ 1 … Δ τ r − 1 Δ τ r .

Furthermore, x ( σ ( t 1 + ) ) − x ( t 1 ) = m 1 x ( t 1 ) , so x ( σ ( t 1 + ) ) = ( 1 + m 1 ) x ( t 1 ) . Then, for any 1 < s ≤ n 0 and σ ( t s − 1 + ) ≤ t < σ ( t s + ) , we can obtain

x ( t ) = ( 1 + ∑ r = 1 ∞ c s r ( t ) ) x σ ( t s − 1 + ) , where x s ( t s − 1 + ) = x ( s ( t s − 1 + ) ) , c s , r ( t ) = ∫ σ ( t s − 1 + ) t f ( τ r ) ∫ σ ( t s − 1 + ) τ r f ( τ r − 1 ) … ∫ σ ( t s − 1 + ) τ 2 f ( τ 1 ) Δ τ 1 … Δ τ r − 1 Δ τ r .

Hence, for t ≥ σ ( t n 0 + ) , by repeating the same iteration process above, we have

x ( t ) = [ 1 + ∑ r = 1 ∞ ∫ σ ( t n 0 + ) t f ( τ r ) ∫ σ ( t n 0 + ) τ r f ( τ r − 1 ) … ∫ σ ( t n 0 + ) τ 2 f ( τ 1 ) Δ τ 1 … Δ τ r − 1 Δ τ r ] x ( σ ( t n 0 + ) ) = [ 1 + ∑ r = 1 ∞ ∫ σ ( t n 0 + ) t f ( τ r ) ∫ σ ( t n 0 + ) τ r f ( τ r − 1 ) … ∫ σ ( t n 0 + ) τ 2 f ( τ 1 ) Δ τ 1 … Δ τ r − 1 Δ τ r ] ∏ s = n 0 1 ( 1 + m s ) [ 1 + ∑ r = 1 ∞ C s , r ] x 0 ,

so the solution of (3.4) given by Lemma 3.3 is obtained. This completes the proof.□

Now, consider the nonhomogeneous linear dynamic equation as follows:

(3.5) { x Δ ( t ) = f ( t ) x ( t ) + h ( t ) x ( t 0 ⁎ ) = x 0 ,

where f , h : T → ℍ , x 0 ∈ ℍ , t 0 ⁎ ∈ T .

Lemma 3.4

The solution of (3.5) is given by

x ( t ) = e f ( t , t 0 ⁎ ) x 0 + ∫ t 0 ⁎ t e f ( t , σ ( τ ) ) h ( τ ) Δ τ .

Moreover, x(t) can be given as

x ( t ) = ( 1 + ∑ n = 1 ∞ ∫ t 0 ⁎ t f ( τ n ) ∫ t 0 ⁎ τ n f ( τ n − 1 ) … ∫ t 0 ⁎ τ 2 f ( τ 1 ) Δ τ 1 … Δ τ n − 1 Δ τ n ) x 0 + ∫ t 0 ⁎ t ( 1 + ∑ n =1 ∞ ∫ σ ( τ ) t f ( τ n ) ∫ σ ( τ ) τ n f ( τ n − 1 ) … ∫ σ ( τ ) τ 2 f ( τ 1 ) Δ τ 1 … Δ τ n − 1 Δ τ n ) h ( τ ) Δ τ .

Proof

For t = t 0 ⁎ , it is obvious that x ( t 0 ⁎ ) = e f ( t 0 ⁎ , t 0 ⁎ ) x 0 + ∫ t 0 ⁎ t 0 ⁎ e f ( t 0 ⁎ , σ ( τ ) ) h ( τ ) Δ τ = x 0 . Moreover,

μ ( t ) x Δ ( t ) = e f ( σ ( t ) , t 0 ⁎ ) x 0 + ∫ t 0 ⁎ σ ( t ) e f ( σ ( t ) , σ ( τ ) ) h ( τ ) Δ τ − e f ( t , t 0 ⁎ ) x 0 + ∫ t 0 ⁎ t e f ( t , σ ( τ ) ) h ( τ ) Δ τ = e f ( σ ( t ) , t 0 ⁎ ) x 0 − e f ( t , t 0 ⁎ ) x 0 + ∫ t 0 ⁎ σ ( t ) e f ( σ ( t ) , σ ( τ ) ) h ( τ ) Δ τ − ∫ t 0 ⁎ t e f ( t , σ ( τ ) ) h ( τ ) Δ τ = μ ( t ) f ( t ) e f ( t , t 0 ⁎ ) x 0 + ∫ t 0 ⁎ σ ( t ) e f ( σ ( t ) , σ ( τ ) ) h ( τ ) Δ τ + ∫ t 0 ⁎ t [ e f ( σ ( t ) , σ ( τ ) ) − e f ( t , σ ( τ ) ) ] h ( τ ) Δ τ = μ ( t ) f ( t ) e f ( t , t 0 ⁎ ) x 0 + μ ( t ) e f ( σ ( t ) , σ ( t ) ) h ( t ) + ∫ t 0 ⁎ t μ ( t ) f ( t ) e f ( t , σ ( τ ) ) h ( τ ) Δ τ = [ f ( t ) x ( t ) + h ( t ) ] μ ( t ) .

Moreover, by Theorem 3.1, we can obtain the desired results. The proof is complete.□

Next, we consider the nonhomogeneous impulsive dynamic equation as follows:

(3.6) { x Δ ( t ) = f ( t ) x ( t ) + h ( t ) , t ≠ t n , Δ ˜ x ( t ) = m n x ( t ) , t = t n , n ∈ Z , x ( t 0 ⁎ ) = x 0 ,

where f , h : T → ℍ , t n ∈ { t 1 , t 2 , … , t n 0 } ∈ [ t 0 ⁎ , t ] , t ∈ [ a , b ] T and 1 + m n ≠ 0 , m n ∈ ℍ .

Theorem 3.2

The solution of (3.6) is given by

Ψ f , h = { ( 1 + ∑ r = 1 ∞ c 1 , r ) x 0 + ∫ t 0 ⁎ t e f ( t , σ ( τ ) ) h ( τ ) Δ τ , t 0 ⁎ ≤ t < σ ( t 1 + ) , ( 1 + ∑ w = 1 ∞ c s , w ) { ∏ v = s − 1 1 ( 1 + m v ) u v + ∑ l = 2 s − 2 ∏ k = s − 1 l + 1 ( 1 + m k ) u k ( 1 + m l ) h l − 1 + ( 1 + m s − 1 ) h s − 2 } + ∫ σ ( t s − 1 + ) t e f ( t , σ ( τ ) ) h ( τ ) Δ τ , σ ( t s − 1 + ) ≤ t < σ ( t s + ) , 1 < s ≤ n 0 , ∏ v = s 1 ( 1 + m v ) u v + ∑ l = 2 s − 1 ∏ k = s l + 1 ( 1 + m k ) u k ( 1 + m l ) h l − 1 + ( 1 + m s ) h s − 1 , t = σ ( t s + ) , 1 < s ≤ n 0 , e f ( t , σ ( t n 0 + ) ) { ∏ v = n 0 1 ( 1 + m v ) u v + ∑ l = 1 n 0 − 1 ∏ k = n 0 l + 1 ( 1 + m k ) u k ( 1 + m l ) h l − 1 + ( 1 + m n 0 ) h n 0 − 1 } + ∫ σ ( t n 0 + ) t e f ( t , σ ( τ ) ) h ( τ ) Δ τ , t > σ ( t n 0 + ) ,

where

u 1 = ( 1 + ∑ n = 1 ∞ C 1 , n ) x 0 + ∫ t 0 ⁎ t 1 e f ( t , σ ( τ ) ) h ( τ ) Δ τ , u v = 1 + ∑ p = 1 ∞ C v , p , 1 < v ≤ n 0 , h l − 1 = ∫ σ ( t l − 1 + ) t l e f ( t , σ ( τ ) ) h ( τ ) Δ τ , 2 ≤ l ≤ n 0 .

Proof

For t 0 ⁎ ≤ t < σ ( t 1 + ) , we have

x ( t 1 ) = [ 1 + ∑ r = 1 ∞ c 1, r ] x 0 + ∫ t 0 ⁎ t 1 e f ( t , σ ( τ ) ) h ( τ ) Δ τ , c 1, r = ∫ t 0 ⁎ t f ( τ r ) ∫ t 0 ⁎ τ r f ( τ r − 1 ) … ∫ t 0 ⁎ τ 2 f ( τ 1 ) Δ τ 1 … Δ τ r − 1 Δ τ r .

Furthermore, x ( σ ( t 1 + ) ) − x ( t 1 ) = m 1 x ( t 1 ) , so x ( σ ( t 1 + ) ) = ( 1 + m 1 ) x ( t 1 ) , for 1 < s ≤ n 0, σ ( t s − 1 + ) ≤ t < σ ( t s + )

x ( t ) = [ 1 + ∑ r = 1 ∞ c s , r ] x σ ( t s − 1 + ) + ∫ σ ( t s − 1 + ) t e f ( t , σ ( τ ) ) h ( τ ) Δ τ = [ 1 + ∑ r =1 ∞ c s , r ] (1 + m s − 1 ) x ( t s − 1 ) + ∫ σ ( t s − 1 + ) t e f ( t , σ ( τ ) ) h ( τ ) Δ τ ,

where

c s , r = ∫ σ ( t s − 1 + ) t f ( τ r ) ∫ σ ( t s − 1 + ) τ r f ( τ r − 1 ) … ∫ σ ( t s − 1 + ) τ 2 f ( τ 1 ) Δ τ 1 … Δ τ r − 1 Δ τ r ,

for 1 < s ≤ n 0, we have

x σ ( t s + ) = ( 1 + m s ) x ( t s ) = ( 1 + m s ) [ ( 1 + ∑ r = 1 ∞ C s , r ) x σ ( t s − 1 + ) + ∫ σ ( t s − 1 ) + t s e f ( t , σ ( τ ) ) h ( τ ) Δ τ ] ,

where

C s , r = ∫ σ ( t s − 1 + ) t s f ( τ r ) ∫ σ ( t s − 1 + ) τ r f ( τ r − 1 ) … ∫ σ ( t s − 1 + ) τ 2 f ( τ 1 ) Δ τ 1 … Δ τ r − 1 Δ τ r .

For t > σ ( t n 0 + ) ,

x ( t ) = [ 1 + ∑ r = 1 ∞ ∫ σ ( t n 0 + ) t f ( τ r ) … ∫ σ ( t n 0 + ) τ 2 f ( τ 1 ) Δ τ 1 … Δ τ r ] x ( σ ( t n 0 + ) ) + ∫ σ ( t n 0 + ) t e f ( t , σ ( τ ) ) h ( τ ) Δ τ .

By repeating the iteration process above, we can obtain

Ψ f , h = { ( 1 + ∑ r = 1 ∞ c 1 , r ) x 0 + ∫ t 0 ⁎ t e f ( t , σ ( τ ) ) h ( τ ) Δ τ , t 0 ⁎ ≤ t ≤ σ ( t 1 + ) , ( 1 + ∑ w = 1 ∞ c s , w ) { ∏ v = s − 1 2 ( 1 + m v ) ( 1 + ∑ p = 1 ∞ C v , p ) × ( 1 + m 1 ) [ ( 1 + ∑ n = 1 ∞ C 1 , n ) x 0 + ∫ t 0 ⁎ t 1 e f ( t , σ ( τ ) ) h ( τ ) Δ τ ] + ∑ l = 2 s − 2 ∏ k = s − 1 l + 1 ( 1 + m k ) ( 1 + ∑ d = 1 ∞ C k , d ) ( 1 + m l ) ∫ σ ( t l − 1 + ) t l e f ( t , σ ( τ ) ) h ( τ ) Δ τ + ( 1 + m s − 1 ) ∫ σ ( t s − 2 + ) t s − 1 e f ( t , σ ( τ ) ) h ( τ ) Δ τ } + ∫ σ ( t s − 1 + ) t e f ( t , σ ( τ ) ) h ( τ ) Δ τ , σ ( t s − 1 + ) ≤ t < σ ( t s + ) , 1 < s ≤ n 0 , ∏ v = s 2 ( 1 + m v ) ( 1 + ∑ p = 1 ∞ C v , p ) ( 1 + m 1 ) [ ( 1 + ∑ n = 1 ∞ C 1 , n ) x 0 + ∫ t 0 ⁎ t 1 e f ( t , σ ( τ ) ) h ( τ ) Δ τ ] + ∑ l = 2 s − 1 ∏ k = s l + 1 ( 1 + m k ) ( 1 + ∑ d = 1 ∞ C k , d ) ( 1 + m l ) ∫ σ ( t l = 1 + ) t l e f ( t , σ ( τ ) ) h ( τ ) Δ τ + ( 1 + m s ) ∫ σ ( t s − 1 + ) t s e f ( t , σ ( τ ) ) h ( τ ) Δ τ , t = σ ( t s + ) , 1 < s ≤ n 0 , [ 1 + ∑ w = 1 ∞ ∫ σ ( t n 0 + ) t f ( τ w ) … ∫ σ ( t n 0 + ) τ 2 f ( τ 1 ) Δ τ 1 … Δ τ w ] × { ∏ v = n 0 2 ( 1 + m v ) ( 1 + ∑ p = 1 ∞ C v , p ) ( 1 + m 1 ) [ ( 1 + ∑ n = 1 ∞ C 1 , n ) x 0 + ∫ t 0 ⁎ t 1 h ( τ ) e f ( t , σ ( τ ) ) Δ τ ] + ∑ l = 2 n 0 − 1 ∏ k = n 0 l + 1 ( 1 + m k ) ( 1 + ∑ d = 1 ∞ C k , d ) ( 1 + m l ) ∫ σ ( t l − 1 + ) t l e f ( t , σ ( τ ) ) h ( τ ) Δ τ + ( 1 + m n 0 ) ∫ σ ( t n 0 − 1 + ) t n 0 e f ( t , σ ( τ ) ) h ( τ ) Δ τ } + ∫ σ ( t n 0 + ) t e f ( t , σ ( τ ) ) h ( τ ) Δ τ , t > t n 0 ,

where

C 1, r = ∫ t 0 ⁎ t 1 f ( τ r ) … ∫ t 0 ⁎ τ 2 f ( τ 1 ) Δ τ 1 … Δ τ r , C s , r = ∫ σ ( t s − 1 + ) t s f ( τ r ) … ∫ σ ( t s − 1 + ) τ 2 f ( τ 1 ) Δ τ 1 … Δ τ r , c 1, r = ∫ t 0 ⁎ t f ( τ r ) … ∫ t 0 ⁎ τ 2 f ( τ 1 ) Δ τ 1 … Δ τ r , c s , r = ∫ σ ( t s − 1 + ) t f ( τ r ) … ∫ σ ( t s − 1 + ) τ 2 f ( τ 1 ) Δ τ 1 … Δ τ r ,

and 1 < s ≤ n 0. The proof is complete.□

Example 3.1

By Theorem 3.2, let T = Z h ˜ , h ˜ > 0 , for 1 < s ≤ n 0, we can get

c s , r = { 0 , r > t − σ ( t s − 1 ) h ˜ , h ˜ r ∏ q ˜ = 1 r f ( t − q ˜ h ˜ ) , r = t − σ ( t s − 1 ) h ˜ , h ˜ r ∏ q ˜ = 1 r − 1 f ( t − q ˜ h ˜ ) ∑ v = 0 t − σ ( t s − 1 ) h ˜ − ( r − 2 ) f ( σ ( t s − 1 ) + v h ˜ ) , r < t − σ ( t s − 1 ) h ˜ ;

C s , r = { 0, r > t s − σ ( t s − 1 ) h ˜ , h ˜ r ∏ v = 1 r f ( t s − v h ˜ ) , r = t s − σ ( t s − 1 ) h ˜ , h ˜ r ∏ q ˜ =1 r − 1 f ( t s − q ˜ h ˜ ) ∑ v =0 t s − σ ( t s − 1 ) h ˜ − ( r − 2) f ( σ ( t s − 1 ) + v h ˜ ) , r < t s − σ ( t s − 1 ) h ˜ ;

c 1, r = { 0 , r > t − t 0 ⁎ h ˜ , h ˜ r ∏ q ˜ =1 r f ( t − q ˜ h ˜ ) , r = t − t 0 ⁎ h ˜ , h ˜ r ∏ q ˜ =1 r − 1 f ( t − q ˜ h ˜ ) ∑ v =0 t − t 0 ⁎ h ˜ − ( r − 2) f ( t 0 ⁎ + v h ˜ ) , r < t − t 0 ⁎ h ˜ ;

C 1 , r = { 0 , r > t 1 − t 0 ⁎ h ˜ , h ˜ r ∏ q ˜ = 1 n f ( t 1 − q ˜ h ˜ ) , r = t 1 − t 0 ⁎ h ˜ , h ˜ r ∏ q ˜ = 1 n − 1 f ( t 1 − q ˜ h ˜ ) ∑ v = 0 t 1 − t 0 ⁎ h − ( r − 2 ) f ( t 0 ⁎ + v h ˜ ) , r < t 1 − t 0 ⁎ h ˜ .

By Theorem 3.2, we can obtain the solution of (3.6) under the time scale Z h ˜ .

Example 3.2

For (3.6), let T = q ℕ 0 , where q > 1, we can get

c s , r = { 0, r > ln t − ln σ ( t s − 1 ) ln q , ( q − 1) r t r q r ( r + 1) 2 ∏ v =1 r f ( t q v ) , r = ln t − ln σ ( t s − 1 ) ln q , ( q − 1) r t r − 1 q r ( r − 1) 2 ∏ v =1 r − 1 f ( t q v ) ∑ l =0 ln t − ln σ ( t s − 1 ) ln q − ( r − 2) f [ σ ( t s − 1 ) q l ] σ ( t s − 1 ) q l , n < ln t − ln σ ( t s − 1 ) ln q ;

C s , r = { 0, r > ln t s − ln σ ( t s − 1 ) ln q , ( q − 1) r t s r q r ( r + 1) 2 ∏ v = 1 r f ( t s q v ) , r = ln t s − ln σ ( t s − 1 ) ln q , ( q − 1) r t s r − 1 q r ( r − 1) 2 ∏ v = 1 r − 1 f ( t s q v ) ∑ l =0 ln t s − ln σ ( t s − 1 ) ln q − ( r − 2) f [ σ ( t s − 1 ) q l ] σ ( t s − 1 ) q l , r < ln t s − ln σ ( t s − 1 ) ln q ;

c 1 , r = { 0 , r > ln t − ln t 0 ⁎ ln q , ( q − 1 ) r t r q r ( r + 1 ) 2 ∏ v = 1 n f ( t q v ) , r = ln t − ln t 0 ⁎ ln q , ( q − 1 ) r t r − 1 q r ( r − 1 ) 2 ∏ v = 1 r − 1 f ( t q v ) ∑ l = 0 ln t − ln t 0 ⁎ ln q − ( r − 2 ) f [ t 0 ⁎ q l ] t 0 ⁎ q l , r < ln t − ln t 0 ⁎ ln q ;

C 1 , r = { 0 , r > ln t 1 − ln t 0 ⁎ ln q , ( q − 1 ) r t 1 r q r ( r + 1 ) 2 ∏ v = 1 r f ( t 1 q v ) , r = ln t 1 − ln t 0 ⁎ ln q , ( q − 1 ) r t 1 r − 1 q r ( r − 1 ) 2 ∏ v = 1 r − 1 f ( t 1 q i ) ∑ k = 0 ln t 1 − ln t 0 ⁎ ln q − ( r − 2 ) f ( t 0 ⁎ q l ) t 0 ⁎ q l , r < ln t 1 − ln t 0 ⁎ ln q .

By Theorem 3.2, we can obtain the solution of (3.6) under the time scale q ℕ 0 .

Let T = ℝ , 1 ≤ s ≤ n 0, r ≥ 1, it easily follows that

c s , r = ∫ t s − 1 t f ( τ r ) ∫ t s − 1 τ r f ( τ r − 1 ) … ∫ t s − 1 τ 2 f ( τ 1 ) d τ 1 … d τ r − 1 d τ r , C s , r = ∫ t s − 1 t s f ( τ r ) ∫ t s − 1 τ r f ( τ r − 1 ) … ∫ t s − 1 τ 2 f ( τ 1 ) d τ 1 … d τ r − 1 d τ r .

Example 3.3

Let T = Z + , consider the following impulsive discrete dynamic equation:

(3.7) { x Δ ( t ) = f ( t ) x ( t ) + h ( t ) , t ≠ 3 n − 1 , Δ ˜ x ( t ) = m n x ( t ) , t = t n = 3 n − 1 , x ( 1 ) = 1 ,

where i, j, and k are the quaternion imaginary units, m n ∈ ℍ , f(t) = 1 + ti, h(t) = 2 + tj, m n = n − 1, Δ ˜ x ( t ) = x ( σ ( t ) ) − x ( t ) .

Proof

For any t ∈ T , t is right-scattered, therefore

x ( 2 ) = x ( 1 ) + ∫ 1 2 [ ( 1 + τ i ) x ( 1 ) + 2 + τ j ] Δ τ = 4 + i + j , x ( t ) = [ 2 + ( t − 1 ) i ] x ( t − 1 ) + 2 + ( t − 1 ) j , t ≠ 3 n , x ( t ) = n x ( t − 1 ) , t = 3 n .

By Theorem 3.2, for n ≥ 1 we can obtain:

u n + 1 = 1 + ∫ 3 n 3 n + 2 ( 1 + τ i ) Δ τ + ∫ 3 n 3 n + 2 ( 1 + τ i ) ∫ 3 n τ ( 1 + η i ) Δ η Δ τ = [2 + (3 n + 1) i ][2 + 3 n i ] , h n = ∫ 3 n 3 n + 2 e f (3 n + 2, σ ( τ ) ) Δ τ = ∫ 3 n 3 n + 1 e f (3 n + 2, σ ( τ ) ) Δ τ + ∫ 3 n + 1 3 n + 2 e f (3 n + 2, σ ( τ ) ) Δ τ = e f (3 n + 2,3 n + 1) h (3 n ) + h (3 n + 1) = [2 + (3 n + 1) i ][2 + 3 n j ] + [2 + (3 n + 1) j ] .

Furthermore, the solution of (3.7) is given by

Ψ f , h = { 1 , t = 1 , 4 + i + j , t = 2 , 3 , ( 2 + 4 i ) x ( 3 ) + 2 + 4 j , t = 4 , ( 2 + 5 i ) x ( 4 ) + 2 + 5 j , t = 5 , n ! ∏ s = n 2 [ 2 + ( 3 s − 2 ) i ] [ 2 + 3 ( s − 1 ) i ] ( 4 + i + j ) + ∑ r = 2 n − 1 n ! ( r − 1 ) ! ∏ s = n r + 1 [ 2 + ( 3 s − 2 ) i ] × [ 2 + 3 ( s − 1 ) i ] { [ 2 + ( 3 r − 2 ) j ] + [ 2 + ( 3 r − 2 ) i ] [ 2 + ( 3 r − 3 ) j ] } + n { [ 2 + ( 3 n − 2 ) j ] + [ 2 + ( 3 n − 2 ) i ] [ 2 + ( 3 n − 3 ) j ] } , t = 3 n , n ≥ 2 [ 2 + 3 n i ] x ( 3 n ) + 2 + 3 n j , t = 3 n + 1 , [ 2 + ( 3 n + 1 ) i ] [ 2 + 3 n i ] x ( 3 n ) + [ 2 + ( 3 n + 1 ) i ] ( 2 + 3 n j ) + 2 + ( 3 n + 1 ) j , t = 3 n + 2 . □

Remark 3.3

Considering the following quaternion ∇-dynamic equations:

{ x ∇ ( t ) = f ( t ) x ( t ) + h ( t ) , t ≠ t n , ∇ x ( t ) = m n x ( t ) , t = t n ,

where f : T → ℍ , m n ∈ ℍ , t n ∈ T , ∇ x ( t ) = x ( t ) − x ρ ( t − ) , the initial value is x ( t 0 ⁎ ) = x 0 , let t n 0 < t n − 1 < … < t 1 < t 0 ⁎ ; thus, the solution Ψ f , h is given by

Ψ f , h { ( 1 + ∑ r = 1 ∞ c 1 , r ) x 0 + ∫ t 0 ⁎ t e f ( t , ρ ( τ ) ) h ( τ ) ∇ τ , ρ ( t 1 − ) < t ≤ t 0 ⁎ , ( 1 + ∑ w = 1 ∞ c s , w ) { ∑ v = s − 1 1 ( 1 − m v ) u v + ∑ l = 2 s − 2 ∏ k = s − 1 l + 1 ( 1 − m k + 1 ) u k ( 1 − m l ) h l − 1 + ( 1 − m s − 1 ) h s − 2 } + ∫ ρ ( t s − 1 − ) t e f ( t , ρ ( τ ) ) h ( τ ) ∇ τ , ρ ( t s − ) < t ≤ ρ ( t s − 1 − ) , 1 < s ≤ n 0 , ∑ v = 1 s ( 1 − m v ) u v + ∑ l = 2 s − 1 ∏ k = s l + 1 ( 1 − m k ) u k ( 1 − m l ) h l − 1 + ( 1 − m s ) h s − 1 , t = ρ ( t s − ) , 1 < s ≤ n 0 , [ 1 + ∑ w = 1 ∞ ∫ ρ ( t n 0 − ) t f ( τ w ) … ∫ ρ ( t n 0 − ) τ 2 f ( τ 1 ) ∇ τ 1 … ∇ τ w ] { ∏ v = n 0 1 ( 1 − m v ) u v + ∑ l = 2 n 0 − 1 ∏ k = n 0 l + 1 ( 1 − m k ) u k × ( 1 − m l ) h l − 1 + ( 1 − m n 0 ) h n 0 − 1 } + ∫ ρ ( t n 0 − ) t e f ( t , ρ ( τ ) ) h ( τ ) ∇ τ , t < ρ ( t n 0 − ) ,

where 1 < s ≤ n 0,

C 1 , r = ∫ t 0 ⁎ t 1 f ( τ r ) … ∫ t 0 ⁎ τ 2 f ( τ 1 ) ∇ τ 1 … ∇ τ r , C s , r = ∫ ρ ( t s − 1 − ) t s f ( τ r ) … ∫ ρ ( t s − 1 − ) τ 2 f ( τ 1 ) ∇ τ 1 … ∇ τ r , c 1 , r = ∫ t 0 ⁎ t f ( τ r ) … ∫ t 0 ⁎ τ 2 f ( τ 1 ) ∇ τ 1 … ∇ τ r , c s , r = ∫ ρ ( t s − 1 − ) t f ( τ r ) … ∫ ρ ( t s − 1 − ) τ 2 f ( τ 1 ) ∇ τ 1 … ∇ τ r , u 1 = ( 1 + ∑ n = 1 ∞ , C 1 , n ) x 0 + ∫ t 0 ⁎ t 1 e f ( t , ρ ( τ ) ) h ( τ ) ∇ τ , u v = 1 + ∑ p = 1 ∞ C v , p , h l − 1 = ∫ ρ ( t l − 1 + ) t l e f ( t , ρ ( τ ) ) h ( τ ) ∇ τ .

4 Quaternion matrix impulsive dynamic equation

Now, consider the impulsive dynamic matrix equation as follows:

(4.1) { X Δ ( t ) = A ( t ) X ( t ) + F ( t ) , t ≠ t h , Δ ˜ X ( t ) = B h X ( t ) , t = t h ,

where A , F : T → ℍ n × n , B h ∈ ℍ , t h ∈ T , Δ ˜ X ( t ) = X ( σ ( t + ) ) − X ( t ) , h ∈ ℕ . The initial value problem

X ( t 0 ⁎ ) = X 0 ∈ ℍ n × n .

Remark 4.1

In (4.1), if t h is the right-dense point, then Δ ˜ X ( t ) = X ( t + ) − X ( t ) ; if t h is the right-scattered point, then Δ ˜ X ( t ) = X ( σ ( t ) ) − X ( t ) .

Lemma 4.1

Let A , B : T → ℍ n × n , Q ( t ) = ∫ t 0 t A ( τ ) ∫ s τ B ( η ) Δ η Δ τ , and t 0 ∈ T be fixed, then Q(t) is differentiable at t with

Q Δ ( t ) = A ( t ) ∫ t 0 t B ( η ) Δ η .

Moreover, if P ( t ) = ∫ t 0 t A ( τ n ) ∫ t 0 τ n A ( τ n − 1 ) … ∫ t 0 τ 2 A ( τ 1 ) Δ τ … Δ τ n − 1 Δ τ n , then

(4.2) P Δ ( t ) = A ( t ) ∫ t 0 t A ( τ n − 1 ) … ∫ t 0 τ 2 A ( τ 1 ) Δ τ 1 … Δ τ n − 1 .

Proof

By Lemma 3.1, we can obtain

[ ∑ s = 1 n ∫ t 0 t a r s ( τ 2 ) ∫ t 0 τ 2 a s l ( τ 1 ) Δ τ 1 Δ τ 2 ] Δ = ∑ s = 1 n a r s ( t ) ∫ t 0 t a s l ( τ 1 ) Δ τ 1 .

For the right-scatted point t ∈ T , the Δ-derivative of F(t) can be calculated as follows:

Q Δ ( t ) = ∫ t 0 σ ( t ) A ( τ ) ∫ s τ B ( η ) Δ η Δ τ − ∫ t 0 t A ( τ ) ∫ s τ B ( η ) Δ η Δ τ μ ( t ) = ∫ t 0 σ ( t ) A ( τ ) Δ τ ∫ s τ B ( η ) Δ η − ∫ t 0 t A ( τ ) Δ τ ∫ s τ B ( η ) Δ η μ ( t ) = [ ∫ t 0 σ ( t ) A ( τ ) Δ τ − ∫ t 0 t A ( τ ) Δ τ ] ∫ s τ B ( η ) Δ η μ ( t ) = μ ( t ) A ( t ) ∫ s t B ( η ) Δ η μ ( t ) = A ( t ) ∫ s t B ( η ) Δ η .

For the right-dense point t ∈ T , the derivative of F(t) is obvious. Similar to the above calculation, by taking P(t) = Q(t), we can get (4.2). The proof is complete.□

Now, we consider the homogenous linear dynamic equation as follows:

(4.3) { X Δ ( t ) = A ( t ) X ( t ), X ( t 0 ⁎ ) = X 0 ,

where A ∈ ℛ r d , X 0 ∈ ℍ n × n , and t 0 ⁎ ∈ T .

Lemma 4.2

For (4.3), if A(t) is uniformly bounded on T , i.e., there exists some constant M A > 0, such that ∥; A ( t ) ∥; ≤ M A for any t ∈ T , then the solution of (4.3) is given as follows:

X ( t ) = ( I + ∑ n = 1 ∞ g n ( t ) ) X 0 ,

where g n ( t ) = ∫ t 0 ⁎ t A ( τ n ) ∫ t 0 ⁎ τ n A ( τ n − 1 ) … ∫ t 0 ⁎ τ 2 A ( τ 1 ) Δ τ 1 … Δ τ n − 1 τ n .

Proof

Let h be constant with h > 0. For t 0 ⁎ ≤ t < t 0 ⁎ + h we have

∥; g n ( t ) ∥; ≤ ∫ t 0 ⁎ t M A ∫ t 0 ⁎ t n M A … ∫ t 0 ⁎ t 2 M A Δ t 1 … Δ t n − 1 Δ t n = M A n h n ( t , t 0 ⁎ ) ≤ M A n ( t − t 0 ⁎ ) n n ! = M A n h n n ! .

By the Weierstrass theorem, the series ∑ n = 1 ∞ M A n h n n ! is convergent (say it is convergent to a ∈ ℝ + ), which implies that the series { ∑ n = 1 ∞ g n ( t ) } is uniformly convergent on T .

Next, we show that the function X ∈ ℛ r d . For the right-dense point t r ∈ T , ∀ ε > 0 , there exists δ ( ε ) = ε / a M f > 0 , such that | t − t r | < δ ( ε ) , we have

∥; X ( t ) − X ( t r ) ∥; = ∥; ∑ n = 1 ∞ ∫ t r t A ( t n ) ∫ t 0 ⁎ t n A ( t n − 1 ) … ∫ t 0 ⁎ t 2 A ( t 1 ) Δ t 1 … Δ t n − 1 Δ t n x 0 ∥; ≤ ∑ n = 1 ∞ M A n | t − t r | ( t − t 0 ⁎ ) n − 1 ( n − 1 ) ! ≤ δ ( ε ) ∑ n = 1 ∞ M A n ( t − t 0 ⁎ ) n − 1 ( n − 1 ) ! ≤ ε a M A M A ∑ n = 1 ∞ M A n − 1 h n − 1 ( n − 1 ) ! < ε .

Thus, X(t r ) is continuous at right-dense. Moreover, since the function A ∈ ℛ r d , it follows that A(t) has the finite left-side limit at a left-dense point. Therefore, X ∈ ℛ r d .

On the other hand, by Lemma 4.1, we can get g n Δ ( t ) = A ( t ) g n − 1 ( t ) , and hence, X ( t ) = ( I + ∑ n = 1 ∞ g n ( t ) ) X 0 is Δ-differentiable with

X Δ ( t ) = ∑ n = 1 ∞ A ( t ) g n − 1 ( t ) X 0 ,

where g 0(t) = I, hence X Δ (t) = A(t)X(t). Therefore, the function series X ( t ) = ( I + ∑ n = 1 ∞ g n ( t ) ) X 0 is a solution of (4.3); according to the continuation theorem of solutions for dynamic equations, X(t) is a solution for (4.3) on T . Furthermore, we assume that X 1 and X 2 are two solutions of (4.3), then

∥; X 1 ( t ) − X 2 ( t ) ∥; ≤ ∫ t 0 t ∥; A ( τ ) ∥; ∥; X 1 ( τ ) − X 2 ( τ ) ∥; Δ τ ≤ M A ∫ t 0 t ∥; X 1 ( τ ) − X 2 ( τ ) ∥; Δ τ .

By Corollary 6.7 from [9] (Bellman inequality on time scale), we can get ∥;X 1(t) − X 2(t)∥; = 0. Therefore, the solution of (4.3) is unique. The proof is complete.□

Theorem 4.1

For (4.3), if for X 0 = I, there exists a unique matrix solution of (4.3), then the generalized exponential function e A ( t , t 0 ⁎ ) can be given by

e A ( t , t 0 ⁎ ) = I + ∑ n = 1 ∞ ∫ t 0 ⁎ t A ( τ n ) ∫ t 0 ⁎ τ n A ( τ n − 1 ) … ∫ t 0 ⁎ τ 2 A ( τ 1 ) Δ τ … Δ τ n − 1 Δ τ n .

Proof

Let

g n = ∫ t 0 ⁎ t A ( τ n ) ∫ t 0 ⁎ τ n A ( τ n − 1 ) … ∫ t 0 ⁎ τ 2 A ( τ 1 ) Δ τ … Δ τ n − 1 Δ τ n ,

for n ≥ 1. By Lemma 4.1, we can obtain

g n Δ = A ( t ) ∫ t 0 ⁎ t A ( τ n − 1 ) … ∫ t 0 ⁎ τ 2 A ( τ 1 ) Δ τ … Δ τ n − 1 = A ( t ) g n − 1 .

By Lemma 4.2, we can obtain the series I + ∑ n = 1 ∞ g n is a unique solution of (4.3) with X ( t 0 ⁎ ) = I . Therefore, by Definition 2.1 we can get

e A ( t , t 0 ⁎ ) = I + ∑ n = 1 ∞ ∫ t 0 ⁎ t A ( τ n ) ∫ t 0 ⁎ τ n A ( τ n − 1 ) … ∫ t 0 ⁎ τ 2 A ( τ 1 ) Δ τ … Δ τ n − 1 Δ τ n .

This completes the proof.□

Now, we consider the following nonhomogeneous linear dynamic equations:

(4.4) { X Δ ( t ) = A ( t ) X ( t ) + F ( t ) , X ( t 0 ⁎ ) = X 0 ,

where A , F ∈ ℛ r d , X 0 ∈ ℍ n × n .

Lemma 4.3

The solution of (4.4) can be given by

X ( t ) = e A ( t , t 0 ⁎ ) X 0 + ∫ t 0 ⁎ t e A ( t , σ ( τ ) ) F ( τ ) Δ τ .

Moreover, X(t) can be given as

X ( t ) = [ I + ∑ n = 1 ∞ ∫ t 0 ⁎ t A ( τ n ) … ∫ t 0 ⁎ τ 2 A ( τ 1 ) Δ τ 1 … Δ τ n ] X 0 + ∫ t 0 ⁎ t [ I + ∑ n = 1 ∞ ∫ σ ( τ ) t A ( τ n ) … ∫ σ ( τ ) τ 2 A ( τ 1 ) Δ τ 1 … Δ τ n ] F ( τ ) Δ τ .

Proof

For t = t 0 ⁎ , it is obvious X ( t 0 ⁎ ) = e A ( t 0 ⁎ , t 0 ⁎ ) X 0 + ∫ t 0 ⁎ t 0 ⁎ e A ( t 0 ⁎ , σ ( τ ) ) F ( τ ) Δ τ = X 0 . A , F ∈ ℛ r d , by Lemma 4.2, we can obtain

X ( t ) = [ I + ∑ n =1 ∞ ∫ t 0 ⁎ t A ( τ n ) ∫ t 0 ⁎ τ 0 A ( τ n − 1 ) … ∫ t 0 ⁎ τ 2 A ( τ 1 ) Δ τ 1 … Δ τ n − 1 Δ τ n ] X 0 + ∫ t 0 ⁎ t e A ( t , σ ( τ ) ) F ( τ ) Δ τ .

is uniformly convergent on T and X ∈ ℛ r d . Moreover,

μ ( t ) X Δ ( t ) = e A ( σ ( t ) , t 0 ⁎ ) x 0 + ∫ t 0 ⁎ σ ( t ) e A ( σ ( t ) , σ ( τ ) ) F ( τ ) Δ τ − e A ( t , t 0 ⁎ ) x 0 + ∫ t 0 ⁎ t e A ( t , σ ( τ ) ) F ( τ ) Δ τ = e A ( σ ( t ) , t 0 ⁎ ) x 0 − e A ( t , t 0 ⁎ ) x 0 + ∫ t 0 ⁎ σ ( t ) e A ( σ ( t ) , σ ( τ ) ) F ( τ ) Δ τ − ∫ t 0 ⁎ t e A ( t , σ ( τ ) ) F ( τ ) Δ τ = μ ( t ) A ( t ) e A ( t , t 0 ⁎ ) x 0 + ∫ t 0 ⁎ σ ( t ) e A ( σ ( t ) , σ ( τ ) ) F ( τ ) Δ τ + ∫ t 0 ⁎ t [ e A ( σ ( t ) , σ ( τ ) ) − e A ( t , σ ( τ ) ) ] F ( τ ) Δ τ = μ ( t ) A ( t ) e A ( t , t 0 ⁎ ) x 0 + μ ( t ) e A ( σ ( t ) , σ ( t ) ) F ( t ) + ∫ t 0 ⁎ t μ ( t ) A ( t ) e A ( t , σ ( τ ) ) h ( τ ) Δ τ = [ A ( t ) x ( t ) + F ( t ) ] μ ( t ) .

Furthermore, by Lemma 4.1, we can obtain the desired results. The proof is complete.□

Theorem 4.2

For (4.1), if any compact interval [ a , b ] T contains only a finite number of points t h , and for all h ∈ ℕ , the matrices I + B h are nonsingular, then the solution of (4.1) can be given by

Ψ A , F = { ( I + ∑ p = 1 ∞ g 0 , p ) X 0 + f 0 , t 0 ⁎ ≤ t < σ ( t 1 + ) , ∏ r = s − 1 0 ( I + B r + 1 ) U r + ∑ v = 1 s − 2 ∏ l = s − 1 v + 1 ( I + B l + 1 ) U l ( I + B v + 1 ) f ( v ) + ( I + B s ) f ( s − 1 ) , t = σ ( t s ) , ( I + ∑ p = 1 ∞ g s , p ) [ ∏ r = s − 1 0 ( I + B r + 1 ) U r + ∑ v = 1 s − 2 ∏ l = s − 1 v + 1 ( I + B l + 1 ) U l ( I + B v + 1 ) f ( v ) + ( I + B s ) f ( s − 1 ) ] + f s , σ ( t s + ) ≤ t < σ ( t s + 1 + ) .

Proof

Let

g s , r = ∫ σ ( t s ) t A ( τ r ) ∫ σ ( t s ) τ r A ( τ r − 1 ) … ∫ σ ( t s ) τ 2 A ( τ 1 ) Δ τ 1 … Δ τ r − 1 Δ τ r , G s , r = ∫ σ ( t s ) t s + 1 A ( τ r ) ∫ σ ( t s ) τ r A ( τ r − 1 ) … ∫ σ ( t s ) τ 2 A ( τ 1 ) Δ τ 1 … Δ τ r − 1 Δ τ r , g 0 , r = ∫ t 0 ⁎ t A ( τ r ) ∫ t 0 ⁎ τ r A ( τ r − 1 ) … ∫ t 0 ⁎ τ 2 A ( τ 1 ) Δ τ 1 … Δ τ r − 1 Δ τ r , G 0 , r = ∫ t 0 ⁎ t 1 A ( τ r ) ∫ t 0 ⁎ τ r A ( τ r − 1 ) … ∫ t 0 ⁎ τ 2 A ( τ 1 ) Δ τ 1 … Δ τ r − 1 Δ τ r ,

where r, s ≥ 1. By Lemmas 4.1, 4.2, and 4.3, we can obtain

{ X ( t ) = ( I + ∑ q = 1 ∞ g q , 0 ) + ∫ t 0 ⁎ t e A ( t , σ ( τ ) ) F ( τ ) Δ τ , t 0 ⁎ ≤ t < σ ( t 1 + ) , X ( t ) = ( I + B s ) X ( t s ) , t = σ ( t s + ) , X ( t ) = ( I + ∑ v = 1 ∞ g s − 1 , v ) X ( σ ( t s − 1 ) ) + ∫ σ ( t s − 1 ) t e A ( t , σ ( τ ) ) F ( τ ) Δ τ , σ ( t s − 1 + ) ≤ t < σ ( t s + ) .

Let

f s = ∫ σ ( t s + ) t e A ( t , σ ( τ ) ) F ( τ ) Δ τ , f ( s ) = ∫ σ ( t s + ) t s + 1 e A ( t , σ ( τ ) ) F ( τ ) Δ τ , s ≥ 1 , f 0 = ∫ t 0 ⁎ t e A ( t , σ ( τ ) ) F ( τ ) Δ τ , f ( 0 ) = ∫ t 0 ⁎ t 1 e A ( t , σ ( τ ) ) F ( τ ) Δ τ , U s = I + ∑ p = 1 ∞ G s , p , s ≥ 1 , U 0 = ( I + ∑ p = 1 ∞ G 0 , p ) X 0 + f ( 0 ) .

Thus, the solution of (4.1) can be given by

Ψ A , F = { ( I + ∑ p = 1 ∞ g 0 , p ) X 0 + f 0 , t 0 ⁎ ≤ t < σ ( t 1 + ) , ∏ r = s − 1 0 ( I + B r + 1 ) U r + ∑ v = 1 s − 2 ∏ l = s − 1 v + 1 ( I + B l + 1 ) U l ( I + B v + 1 ) f ( v ) + ( I + B s ) f ( s − 1 ) , t = σ ( t s ) , ( I + ∑ p = 1 ∞ g s , p ) [ ∏ r = s − 1 0 ( I + B r + 1 ) U r + ∑ v = 1 s − 2 ∏ l = s − 1 v + 1 ( I + B l + 1 ) U l ( I + B v + 1 ) f ( v ) + ( I + B s ) f ( s − 1 ) ] + f s , σ ( t s + ) ≤ t < σ ( t s + 1 + ) .

The proof is complete.□

Remark 4.2

The Cauchy matrix of (4.1) is as follows:

W ( t , z ) = { U ( t , z ) + F F ( t , z ) , t , z ∈ ( t s − 1 , t s ] , U ( t , σ ( t s + ) ) ( I + B s ) [ U ( t s , z ) + F F ( t s , z ) ] + F F ( t , σ ( t s + ) ) , t s − 1 < z ≤ t s < t ≤ t s + 1 , U ( t , t s + ) ( I + B s ) − 1 [ U ( σ ( t s + ) , z ) + F F ( σ ( t s + ) , z ) ] + F F ( t , t s ) , t s − 1 < t ≤ t s < z ≤ t s + 1 , U ( t , σ ( t s + ) ) { ∏ l = s s 1 + 1 ( I + B l ) U ( t l , σ ( t l − 1 + ) ) ( I + B s 1 ) [ U ( t s 1 , z ) + F F ( t s 1 , z ) ] + ∑ v = s 1 + 1 s − 1 ∏ l = s v + 1 ( I + B l ) U ( t l , σ ( t l − 1 + ) ) ( I + B v ) F F ( t v , σ ( t v − 1 ) ) + ( I + B s ) F F ( t s , σ ( t s − 1 ) ) } + F F ( t , σ ( t s + ) ) , t s 1 − 1 < z ≤ t s 1 < t s < t ≤ t s + 1 , U ( t , t s 1 ) { ∏ l = s 1 s − 1 ( I + B v ) − 1 U ( σ ( t v + ) , t v + 1 ) ( I + B s ) − 1 [ U ( σ ( t s + ) , z ) + F F ( σ ( t s + ) , z ) ] + ∑ v = s 1 + 1 s − 1 ∏ l = s 1 v ( I + B l ) − 1 U ( σ ( t l + ) , t l + 1 ) ( I + B v ) − 1 F F ( σ ( t v + ) , t v + 1 ) + ( I + B s 1 ) F F ( σ ( t s 1 + ) , t s 1 + 1 ) } + F F ( t , t s 1 ) , t s 1 − 1 < t ≤ t s 1 < t s < z ≤ t s + 1 ,

where U ( t , z ) = [ I + ∑ n = 1 ∞ ∫ z t A ( τ n ) … ∫ z τ 2 A ( τ 1 ) Δ τ 1 … Δ τ n ] , F F ( t , z ) = ∫ z t e A ( t , σ ( τ ) ) F ( τ ) Δ τ , that is

W ( t , t ) = I , W ( σ ( t s + ) , z ) = [ I + B s ] W ( t s , s ) , W ( z , σ ( t s + ) ) = W ( z , t s ) [ I + B s ] − 1 , W Δ ( t , z ) = A ( t ) X ( t ) X ( σ ( t s + ) ) + F ( t ) , t ∈ [ σ ( t s + ) , t s + 1 ] .

Remark 4.3

The system

(4.5) { X Δ ( t ) = A ( t ) X ( t ) + F ( t ) , t ≠ t h , Δ ˜ X ( t ) = X ( t ) B h , t = t h ,

where A , F : T → ℍ n × n , B h ∈ ℍ , t h ∈ T , Δ ˜ X ( t ) = X ( σ ( t + ) ) − X ( t ) , h ∈ ℕ . The initial value is X ( t 0 ⁎ ) = X 0 ∈ ℍ . For t < t 0 ⁎ , the solution of (4.5) can be given by

Ψ A , F = { ( I + ∑ p = 1 ∞ g 0 , p ) X 0 + f 0 , t 0 ⁎ ≤ t < σ ( t 1 + ) , ∏ l = 1 s U s − l ∏ v = 1 s ( I + B v ) + ∑ p = 0 s − 2 ∏ w = s − 1 p + 1 U w f ( p ) ∏ r = p + 1 s ( I + B r ) + f ( s − 1 ) ( I + B s ) , t = σ ( t s ) , ( 1 + ∑ n = 1 ∞ g s , n ) [ ∏ l = 1 s U s − l ∏ v = 1 s ( I + B v ) + ∑ p = 0 s − 2 ∏ w = s − 1 p + 1 U w f ( p ) ∏ r = p + 1 s ( I + B r ) + f ( s − 1 ) ( I + B s ) ] + ∫ σ ( t s ) t F ( τ ) Δ τ , σ ( t s + ) ≤ t < σ ( t s + 1 ) ,

where g s,n , U s , f 0, and f (r) are defined in Theorem 4.2.

Example 4.1

For the system (4.1), when T = Z h ˜ , h ˜ > 0 , for s ≥ 1, we obtain

g s , r = { 0 , r > t − σ ( t s ) h ˜ , h ˜ r ∏ q ˜ =1 r A ( t − q ˜ h ˜ ) , r = t − σ ( t s ) h ˜ , h ˜ r ∏ q ˜ =1 r − 1 A ( t − q ˜ h ˜ ) ∑ v =0 t − σ ( t s ) h ˜ − ( r − 2) A ( σ ( t s ) + v h ˜ ) , r < t − σ ( t s ) h ˜ ;

G s , r = { 0 , r > t s + 1 − σ ( t s ) h ˜ , h ˜ r ∏ v = 1 r A ( t s + 1 − v h ˜ ) , r = t s + 1 − σ ( t s ) h ˜ , h ˜ r ∏ q ˜ =1 r − 1 A ( t s + 1 − q ˜ h ˜ ) ∑ v =0 t s + 1 − σ ( t s ) h ˜ − ( r − 2) A ( σ ( t s ) + v h ˜ ) , r < t s + 1 − σ ( t s ) h ˜ ;

g 0, r = { 0, r > t − t 0 ⁎ h ˜ , h ˜ r ∏ q ˜ =1 r A ( t − q ˜ h ˜ ) , r = t − t 0 ⁎ h ˜ , h ˜ r ∏ q ˜ =1 r = 1 A ( t − q ˜ h ˜ ) ∑ v =0 t − t 0 ⁎ h ˜ − ( r − 2) A ( t 0 ⁎ + v h ˜ ) , r < t − t 0 ⁎ h ˜ ;

G 0 , r = { 0 , r > t 1 − t 0 ⁎ h ˜ , h ˜ r ∏ q ˜ = 1 n A ( t 1 − q ˜ h ˜ ) , r = t 1 − t 0 ⁎ h ˜ , h ˜ r ∏ q ˜ = 1 n − 1 A ( t 1 − q ˜ h ˜ ) ∑ v = 0 t 1 − t 0 h ˜ − ( r − 2 ) A ( t 0 + v h ˜ ) , r < t 1 − t 0 ⁎ h ˜ .

By Theorem 4.2, we can obtain the solution of (4.1) on the time scale Z h ˜ .

Example 4.2

For the system (4.1), when T = q ℕ 0 , where q > 1, for r , s ∈ ℕ we can get

g s , r = { 0 , r > ln t − ln σ ( t s ) ln q , ( q − 1 ) r t r q r ( r + 1 ) 2 ∏ v = 1 r A ( t q v ) , r = ln t − ln σ ( t s ) ln q , ( q − 1 ) r t r − 1 q r ( r − 1 ) 2 ∏ v = 1 r − 1 A ( t q v ) ∑ l = 0 ln t − ln σ ( t s ) ln q − ( r − 2 ) A ( σ ( t s ) q l ) σ ( t s ) q l , n < ln t − ln σ ( t s ) ln q ;

G s , r = { 0 , r > ln t s + 1 − ln σ ( t s ) ln q , ( q − 1 ) r t s + 1 r q r ( r + 1 ) 2 ∏ v = 1 r A ( t s + 1 q v ) , r = ln t s + 1 − ln σ ( t s ) ln q , ( q − 1 ) r t s + 1 r − 1 q r ( r − 1 ) 2 ∏ v = 1 r − 1 A ( t m + 1 q v ) ∑ l = 0 ln t s + 1 − ln σ ( t s ) ln q − ( r − 2 ) A ( σ ( t s ) q l ) σ ( t s ) q l , r < ln t s + 1 − ln σ ( t s ) ln q ;

g 0 , r = { 0 , r > ln t − ln t 0 ⁎ ln q , ( q − 1 ) r t r q r ( r + 1 ) 2 ∏ v = 1 n A ( t q v ) , r = ln t − ln t 0 ⁎ ln q , ( q − 1 ) r t r − 1 q r ( r − 1 ) 2 ∏ v = 1 r − 1 A ( t q v ) ∑ 0 ln t − ln t 0 ⁎ ln q − ( r − 2 ) A ( t 0 ⁎ q l ) t 0 ⁎ q l , r < ln t − ln t 0 ⁎ ln q ;

G 0 , r = { 0 , r > ln t 1 − ln t 0 ⁎ ln q , ( q − 1 ) r t 1 r q r ( r + 1 ) 2 ∏ v = 1 r A ( t 1 q v ) , r = ln t 1 − ln t 0 ⁎ ln q , ( q − 1 ) r t 1 r − 1 q r ( r − 1 ) 2 ∏ v = 1 r − 1 A ( t 1 q i ) ∑ k = 0 ln t 1 − ln t 0 ⁎ ln q − ( r − 2 ) A ( t 0 ⁎ q l ) t 0 ⁎ q l , r < ln t 1 − ln t 0 ⁎ ln q .

By Theorem 4.2, we can obtain the solution of (4.1) on the time scale q ℕ 0 .

Let T = ℝ , s ≥ 0, r ≥ 1, we can obtain

g s , r = ∫ t s t A ( τ r ) ∫ t s τ r A ( τ r − 1 ) … ∫ t s τ 2 A ( τ 1 ) d τ 1 … d τ r − 1 d τ r , G s , r = ∫ t s t s + 1 A ( τ r ) ∫ t s τ r A ( τ r − 1 ) … ∫ t s τ 2 A ( τ 1 ) d τ 1 … d τ r − 1 d τ r .

The following theorem can be obtained immediately by Remark 2.3 and Theorem 4.2.

Theorem 4.3

If Ψ A , F ∈ D , then the Liouville formula of the quaternion impulsive dynamic equations (4.1) can be given by ddet Ψ A,F .

Proof

By Definition 2.4, Remark 2.3, and Theorem 4.2, the result is obvious.□

Theorem 4.4

Let X(·) = [x rh (·)] n×n , x r h : T → ℍ , where 1 ≤ r, h ≤ n, and X(t) is a solution of (4.1). If A(t), F(t), B n , and X 0 are diagonal matrices, then the Liouville formula of (4.1) can be given as

ddet X ( t ) = ∏ v = 1 n ℌ x v v .

Proof

For any x : T → ℍ , we can obtain ℌ ( x ( ⋅ ) ) ∈ ℝ . On the other hand, if A(t), F(t), B n , and X 0 are diagonal matrices, then the solution X(t) is a diagonal matrix. Hence, we can obtain

ℌ ( X ( t ) ) = ℭ 1 ( X ( t ) ) ℭ 1 ( X ( t ) ) ¯ + ℭ 2 ( X ( t ) ) ℭ 2 ( X ( t ) ) ¯ = [ ℌ ( x r h ( t ) ) ] n × n ,

where 1 ≤ r, h ≤ n, for r ≠ h, ℌ ( x r h ( t ) ) = 0 ; for r = h, ℌ ( x r h ( t ) ) ∈ ℝ . Therefore,

ddet X ( t ) = ∏ v = 1 n ℌ x v v .

This completes the proof.□

Example 4.3

Let T = ℕ , consider the Liouville formula of the following dynamic equations:

(4.6) { X Δ ( t ) = A ( t ) x ( t ) + F ( t ) , t ≠ 3 n , Δ ˜ X ( t ) = B n ( t ) , t = 3 n , X ( 1 ) = X 0 ,

where

A ( t ) = F ( t ) = [ t + 1 + ( − 1 ) t 2 + j ( − 1 ) t 0 0 t + 1 + ( − 1 ) t 2 + i ( − 1 ) t ] ,

B n = [ − 1 + t 3 1 0 − 1 + t 3 ] , X 0 = [ 2 0 0 1 ] .

By Theorem 4.2, for s ≥ 0, we can get

G s , 1 = A ( 3 s + 1 ) + A ( 3 s + 2 ) , G s , 2 = A ( 3 s + 2 ) A ( 3 s + 1 ) , G s , r = 0 , r ≥ 3 .

Hence, we have

U s = I + A ( 3 s + 1 ) + A ( 3 s + 2 ) + A ( 3 s + 2 ) A ( 3 s + 1 ) = [ I + A ( 3 s + 2 ) ] [ I + A ( 3 s + 1 ) ] ∈ ℝ 2 × 2 .

Moreover,

f ( s ) = ∫ 3 s + 1 3 s + 3 e A ( 3 s + 3 , σ ( τ ) ) Δ τ = ∫ 3 s + 1 3 s + 2 e A ( 3 s + 3 , σ ( τ ) ) Δ τ + ∫ 3 s + 2 3 s + 3 e A ( 3 s + 3 , σ ( τ ) ) Δ τ = e A ( 3 s + 3 , 3 s + 2 ) F ( 3 s + 1 ) + F ( 3 s + 2 ) = [ I + A ( 3 s + 2 ) ] F ( 3 s + 1 ) + F ( 3 s + 2 ) = [ I + A ( 3 s + 2 ) ] [ I + A ( 3 s + 1 ) ] − I = U s − I .

For U s ∈ ℝ 2 × 2 , we have f ( s ) ∈ ℝ 2 × 2 . By Theorem 4.2, when t = σ ( t s + ) , we can obtain

X ( t ) = ∏ r = s − 1 0 ( I + B r + 1 ) U r + ∑ v = 1 s − 2 ∏ l = s − 1 v + 1 ( I + B l + 1 ) U l ( I + B v + 1 ) f ( v ) + ( I + B s ) f ( s − 1 ) .

Thus, X ∈ ℝ 2 × 2 , hence for t = t A , where t A ∈ { 1 , 3 n , 3 n + 1 : n ∈ ℕ } , the Liouville formula of (4.6) can be given by ddet X(t).

Example 4.4

Let T = q ℕ , for q > 1, consider the Liouville formula of the impulsive dynamic equations as follows:

(4.7) { X Δ ( t ) = A ( t ) X ( t ) + F ( t ) , t ≠ t n = q 3 n − 1 , Δ ˜ X ( t ) = B n ( t ) , t = t n = q 3 n − 1 , X (2)= X 0 ,

where

A ( t ) = F ( t ) = 1 t [ 1 + ( − 1 ) ln t ln q t q 1 − ( − 1 ) ln t ln q 2 j 0 0 − 2 + ( − 1 ) ln t ln q t q 1 − ( − 1 ) ln t ln q 2 i ] ,

B n = [ t 3 t t 2 t + 3 ] , X 0 = [ 3 0 0 4 ] .

By the definition of the function matrix A(t), we can obtain

A ( q 3 n ) = q − 3 n [ 1 + ( − 1 ) 3 n q 3 n q 1 − ( − 1 ) 3 n 2 j 0 0 − 2 + ( − 1 ) 3 n q 3 n q 1 − ( − 1 ) 3 n 2 i ] ,

A ( q 3 n + 1 ) = q − 3 n − 1 [ 1 + ( − 1 ) 3 n + 1 q 3 n + 1 q 1 − ( − 1 ) 3 n + 1 2 j 0 0 − 2 + ( − 1 ) 3 n + 1 q 3 n + 1 q 1 − ( − 1 ) 3 n + 1 2 i ] .

Therefore, we can get q 3 n A ( q 3 n ) + q 3 n + 1 A ( q 3 n + 1 ) ∈ ℝ 2 × 2 . Hence, for any n ∈ ℕ , we have

U n = I + ∫ q 3 n q 3 n + 2 A ( τ ) Δ τ + ∫ q 3 n q 3 n + 2 A ( τ ) ∫ q 3 n τ A ( τ 1 ) Δ τ 1 Δ τ = I + A ( q 3 n ) μ ( q 3 n ) + A ( q 3 n ) μ ( q 3 n + 1 ) + A ( q 3 n + 1 ) μ ( q 3 n + 1 ) A ( q 3 n ) μ ( q 3 n ) = [ I + A ( q 3 n + 1 ) μ ( q 3 n + 1 ) ] [ I + A ( q 3 n ) μ ( q 3 n ) ] .

Thus, U n ∈ ℝ 2 × 2 . On the other hand,

f ( n ) = ∫ q 3 n q 3 n + 2 e A ( q 3 n + 2 , σ ( τ ) ) F ( τ ) Δ τ = [ I + A ( q 3 n + 1 ) μ ( q 3 n + 1 ) ] F ( q 3 n ) μ ( q 3 n ) + F ( q 3 n + 1 ) μ ( q 3 n + 1 ) = U n − I ∈ ℝ 2 × 2 .

Therefore, U n ∈ ℝ 2 × 2 , and for (4.7), we can obtain B n , f ( n ) , X 0 ∈ ℝ 2 × 2 . By Theorem 4.2, when t = σ(t n ), we have

X ( t ) = ∏ r = s − 1 0 ( I + B r + 1 ) U r + ∑ v = 1 s − 2 ∏ l = s − 1 v + 1 ( I + B l + 1 ) U l ( I + B v + 1 ) f ( v ) + ( I + B s ) f ( s − 1 ) .

Hence, for t = t A , where t A ∈ {2,t n ,σ(t n )}, we have X ∈ ℝ 2 × 2 ; thus, the Liouville formula of (4.7) can be given by ddet X(t) = det X(t) det X(t).

  1. Competing interests: The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the Youth Fund of NSFC (No. 11961077, 11601470), IRTSTYN, and Joint Key Project of Yunnan Provincial Science and Technology Department of Yunnan University (No. 2018FY001(-014)).

References

[1] W. R. Hamilton, R. Dimitrid, and B. Goldsmith, The mathematical tourist, Math. Intelligencer 11 (1989), 29–30.10.1007/BF03023819Search in Google Scholar

[2] A. Handson and H. Hui, Quaternion frame approach to streamline visualization, IEEE Trans. Vis. Comput. Graph. 1 (1995), 164–172.10.1109/2945.468403Search in Google Scholar

[3] K. I. Kou and Y. H. Xia, Linear quaternion differential equations: basic theory and fundamental results, Stud. Appl. Math. 141 (2018), 3–45.10.1111/sapm.12211Search in Google Scholar

[4] Z. Cai and K. I. Kou, Laplace transform: a new approach in solving linear quaternion differential equations, Math. Meth. Appl. Sci. 41 (2018), 4033–4048.10.1002/mma.4415Search in Google Scholar

[5] J. Zhu and J. Sun, Existence and uniqueness results for quaternion-valued nonlinear impulsive differential systems, J. Syst. Sci. Compl. 31 (2018), 596–607.10.1007/s11424-017-6158-9Search in Google Scholar

[6] J. Zhu and J. Sun, Global exponential stability of Clifford-valued recurrent neural networks, Neurocomputing 173 (2016), 685–689.10.1016/j.neucom.2015.08.016Search in Google Scholar

[7] K. Kou, W. Liu, and Y. Xia, Solve the linear quaternion-valued differential equations having multiple eigenvalues, J. Math. Phys. 60 (2019), 10.1063/1.5040237.Search in Google Scholar

[8] D. Cheng, K. I. Kou, and Y. H. Xia, A unified analysis of linear quaternion dynamic equations on time scales, J. Appl. Anal. Comput. 8 (2018), 172–201.10.11948/2018.172Search in Google Scholar

[9] M. Bohner and A. Peterson, Dynamic Equations on Time Scales – An Introduction with Applications, Birkhauser, Boston, 2001.10.1007/978-1-4612-0201-1Search in Google Scholar

[10] R. P. Agarwal, M. Bohner, and P. J. Wong, Sturm-Liouville eigenvalue problems on time scales, Appl. Math. Comput. 99 (1999), 153–166.10.1016/S0096-3003(98)00004-6Search in Google Scholar

[11] C. Wang and R. P. Agarwal, Almost periodic solution for a new type of neutral impulsive stochastic Lasota-Wazewska timescale model, Appl. Math. Lett. 70 (2017), 58–65.10.1016/j.aml.2017.03.009Search in Google Scholar

[12] C. Wang, R. P. Agarwal, and R. Sakthivel, Almost periodic oscillations for delay impulsive stochastic Nicholson’s blowflies timescale model, Comput. Appl. Math. 37 (2018), 3005–3026.10.1007/s40314-017-0495-0Search in Google Scholar

[13] C. Wang and R. Sakthivel, Double almost periodicity for high-order Hopfield neural networks with slight vibration in time variables, Neurocomputing 282 (2018), 1–15.10.1016/j.neucom.2017.12.008Search in Google Scholar

[14] Q. Kong and A. Zafer, Lower bounds for the eigenvalues of first-order nonlinear Hamiltonian systems on time scales, Appl. Math. Lett. 90 (2019), 154–161.10.1016/j.aml.2018.10.027Search in Google Scholar

[15] G. T. Stamov and J. O. Alzabut, Almost periodic solutions for abstract impulsive differential equations, Nonlinear Anal. 72 (2010), 2457–2464.10.1016/j.na.2009.10.042Search in Google Scholar

[16] I. M. Stamova and G. T. Stamov, Mittag-Leffler synchronization of fractional neural networks with time-varying delays and reaction-diffusion terms using impulsive and linear controllers, Neural Networks 96 (2017), 22–32.10.1016/j.neunet.2017.08.009Search in Google Scholar PubMed

[17] I. M. Stamova, Impulsive control for stability of n-species Lotka-Volterra cooperation models with finite delays, Appl. Math. Lett. 23 (2010), 1003–1007.10.1016/j.aml.2010.04.026Search in Google Scholar

[18] V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.10.1142/0906Search in Google Scholar

[19] H. Aslaksen, Quaternionic determinants, Math. Intell. 18 (1996), 57–65.10.1007/978-1-4613-0195-0_13Search in Google Scholar
Received: 2019-09-06
Accepted: 2020-01-20
Published Online: 2020-05-26

© 2020 Zhien Li and Chao Wang, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
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  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
https://doi.org/10.1515/math-2020-0021
Keywords for this article
quaternion dynamic equations; Cauchy matrix; Liouville formula; impulsive; time scales
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  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
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  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
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  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
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  98. Inequalities for the generalized trigonometric and hyperbolic functions
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  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
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  107. Approximation properties of tensor norms and operator ideals for Banach spaces
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  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
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  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
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  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
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  123. Special Issue on Graph Theory (GWGT 2019), Part II
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  129. Bipartite graphs with close domination and k-domination numbers
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  134. Special Issue on Variational/Hemivariational Inequalities
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  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
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  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
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  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
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  148. Quasilinear Dirichlet problems with competing operators and convection
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  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
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