The Non-Eigenvalue Form of Liouville’s Formula and α-Matrix Exponential Solutions for Combined Matrix Dynamic Equations on Time Scales

In this paper, the non-eigenvalue forms of Liouville’s formulas for delta, nabla and α -diamond matrix dynamic equations on time scales are given and proved. Meanwhile, a diamond matrix exponential function (or α -matrix exponential function) is introduced and some classes of homogenous linear diamond- α dynamic equations which possess the α -matrix exponential solutions is studied. The difference and relation of non-eigenvalue forms of Liouville’s formulas among these representative types of dynamic equations is investigated. Moreover, we establish some sufficient conditions to guarantee transformational relation of Liouville’s formulas and exponential solutions among these types of matrix dynamic equations. In addition, we provide several examples on various time scales to illustrate the effectiveness of our result.


Introduction
As an effective and powerful tool, time scale calculus is initiated to unify continuous and discrete analysis and is extensively applied to study dynamic equations [1][2][3][4][5]. Liouville's formula and Liouville's problems are very important topic in ordinary differential equations [6][7][8][9][10]. In Theorem 5.28 from [9], Liouville's formula of dynamic equations on time scales is given by Bohner and Peterson. The theorem is given as follows: Theorem 1. Let A ∈ R be a 2 × 2-matrix-valued function and assume that X is a solution of X ∆ = A(t)X. Then X satisfies Liouville's formula det X(t) = e trA+µ det A (t, t 0 ) det X(t 0 ), for t ∈ T. (1) Please note that Liouville's formula given by the form (1) is very convenient to use due to the main reason that trA + µ det A is independent of the eigenpolynomial and eigenvalue of A. The Liouville's formula for n × n-matrix dynamic equations on time scales were studied in [11,12]. In [11,12], the authors provided the nice form of Liouville's formula by considering the eigenpolynomial and eigenvalue of the coefficient matrix of the dynamic equations. However, if A is a n × n-matrix-valued function for n sufficiently large, the calculation of eigenpolynomial and eigenvalue of A becomes a complicated task and cannot be always achieved, so it will be a better way to provide a matrix form of Liouville's formula for the case of A : T → R n×n to avoid the calculation of eigenpolynomial and eigenvalue of A, similar to (1). Unfortunately, for n ≥ 3, the matrix X that is a solution of dynamic equations X ∆ = A(t)X does not satisfy (1), i.e., the nice form (1) of Liouville's formula will not hold for n ≥ 3.
On the other hand, the combined dynamic derivatives on time scales was proposed in [13], which can unify ∆-dynamic derivative (α = 1) and ∇-dynamic derivative (α = 0). Moreover, this combined type of derivative includes the hybrid dynamic derivatives between ∆ and ∇ cases and was used to study various complex dynamic equations and inequalities on time scales [14][15][16][17][18].
However, since there is no Liouville's formula of diamond-α matrix dynamic equations, in this paper, we make the following contributions: (i) the combined matrix exponential function is introduced and studied; (ii) Liouville's formula of diamond-α matrix dynamic equations is obtained without considering the eigenpolynomial and eigenvalue; (iii) some classes of diamond-α matrix dynamic equations which have α-matrix exponential solutions are investigated; (iv) the obtained results are completely new even for ∆ and ∇-matrix dynamic equations and several examples on various time scales are provided.

Liouville'S Formula for ∆-dynamic Equations
In this section, we will derive the non-eigenvalue form of Liouville's formula for ∆-matrix dynamic equations which will be used to study the combined dynamic equations on time scales. Definition 1 ( [9]). Define the forward jump operator σ : T → T by σ(t) = inf{s ∈ T : s > t}; the backward jump operator ρ : T → T by ρ(t) = sup{s ∈ T : s < t}; the graininess function µ : T → [0, +∞) by µ(t) = σ(t) − t ( denoted by µ ); the ν(t) by ν(t) = t − ρ(t) ( denoted by ν ). Definition 2 ([9]). Definite e A (t, t 0 ) by the unique matrix solution of the initial value problem: Lemma 1. Let A be an upper triangular n × n-matrix-valued function, then A is regressive iff each diagonal element of A is regressive.

Proof. Let
Therefore A is regressive iff each diagonal element of A is regressive.
Remark 1. Let A : T → R n×n be an upper triangular matrix, we can obtain Hence for n > 2, which implies that I + µA is invertible does not be equivalent to trA + µ det A is regressive, i.e., Liouville's Formula (1) is not suitable for n ≥ 3.
It is easy to check the following ∆-derivative formula of determinant function by using the determinant algorithm and Theorem 1.20 from [9]. Lemma 2. Let A : T → R n×n be the following function matrix: Remark 2. In Lemma 2, notice that a ij denotes the element that is located in the ith row and the jth column, i.e., 1 ≤ i, j ≤ n. Hence A ∆ (t) equals to the following: where 3 ≤ i ≤ n − 1. For convenience, we denote the sum by (3). Similarly, the determinant symbol is with the same meaning for all theorems, lemmas and examples.
Theorem 2 (Liouville's formula). Let A ∈ R be an upper triangular n × n-matrix-valued function and assume that X is a solution of X ∆ = A(t)X. Then X satisfies Liouville's formula Proof. For the matrix A, by Lemma 1, we can obtain A is regressive iff each diagonal element of A is regressive. Let This completes the proof.
Hence A is regressive iff the scalar-valued function trA + µdetA is regressive.

Remark 4.
Assume that for n = k, we can obtain similar characterizations of Remark 3. For n = k + 1, let where A k+1 is (k + 1) × (k + 1)-matrix-valued function, A k is k × k-matrix-valued function. Then Hence there exists A k+1 such that A k+1 is regressive and so is trA k+1 + µ det A k+1 .
Theorem 3 (Liouville's formula). Let A ∈ R be n × n-matrix-valued matrix function and assume that X is a solution of X ∆ = A(t)X. Then X satisfies Liouville's formula Proof. For n = 2, by Remark 3, A is regressive implies trA + µ det A is regressive. Let For n ≥ 3, let Hence by Lemma 2, we have Then This completes the proof.
Theorem 4. Let A be a n × n-matrix-valued function. Then A is regressive iff the scalar-valued function Proof. By Theorem 3, for n = 2 we can obtain Next, let Therefore A is regressive iff the scalar-valued function det A i is regressive. This completes the proof.

Liouville'S Formula for ∇-Dynamic Equations and Some Lemmas
In the following, we will obtain Liouville's formula for ∇-dynamic equations and some lemmas which will be used to discuss α-diamond dynamic equations are established.

Lemma 3. By Definition 2, the matrix function e
On the other hand, by The proof is complete.
Next, we consider the dynamic equations by nabla derivative.

Lemma 4.
Let A be an upper triangular n × n-matrix-valued function, then A is regressive iff each diagonal element of A is regressive.

Proof. Let
Therefore A is regressive iff each diagonal element of A is regressive.

Remark 5.
Let A : T → R n×n be an upper triangular matrix, we can obtain Hence for n > 2, which implies that I + νA is invertible does not be equivalent to trA + ν det A is regressive.
It is easy to check the following ∇-derivative formula of determinant function.

Lemma 5.
Let A : T → R n×n be the following function matrix:   Theorem 5 (Liouville's formula). Let A ∈ R be an upper triangular n × n-matrix-valued function and assume that X is a solution of X ∇ = A(t)X. Then X satisfies Liouville's formula This completes the proof.
Hence A is regressive iff the scalar-valued function trA − ν det A is regressive.

Remark 7.
Assume that for n = k, we can obtain similar characterizations of Remark 6. For n = k + 1, let where A k+1 is (k + 1) × (k + 1)-matrix-valued function, A k is k × k-matrix-valued function. Then Hence there exists A k+1 such that A k+1 is regressive and so is trA k+1 − ν det A k+1 .
Theorem 6 (Liouville's formula). Let A ∈ R be n × n-matrix-valued matrix function and assume that X is a solution of X ∇ = A(t)X. Then X satisfies Liouville's formula Proof. For n = 2, by Remark 4, A is regressive implies trA − ν det A is regressive. Let A(t) = a 11 a 12 a 21 a 22 , X(t) = x 11 x 12 x 21 x 22 .
Then det X(t) For n ≥ 3, let  x n1 x n2 . . . Then This completes the proof.

Theorem 7.
Let A be a n × n-matrix-valued function. Then A is regressive iff the scalar-valued function n ∑ i=1 detÃ i is regressive, whereÃ i is defined by Theorem 6.
Proof. By Theorem 6, for n = 2 we can obtain Next, let function is regressive iff the scalar-valued function Therefore A is regressive iff the scalar-valued function detÃ i is regressive. This completes the proof.

Definition 3 ([19]
). Defineê A (t, t 0 ) by the unique matrix solution of the initial value problem: Lemma 6. By Definition 3, the matrix functionê A (t, t 0 ) is delta differentiable at t witĥ On the other hand, by f (σ(t)) − f (t) = f ∆ (t)µ(t), we can obtain The proof is complete. (4), let T = R, Liouville's formula can be given as:

Example 4. For
In fact, let T = R, we can obtain ν(t) = 0 for any t ∈ T, hence X ∇ (t) = X . The result is obvious.

Liouville'S Formula of Diamond-α Dynamic Equations
In the sequel, we will introduce the α-matrix exponential function and obtain Liouville's formula of diamond-α dynamic equations. Several examples are provided on various time scales.

Definition 4 ([13]
). Let T be a time scale and f (t) be differentiable on T in the ∆ and ∇ sense. For t ∈ T we define the diamond-α dynamic derivative f ♦ α (t) by Thus f is diamond-α differentiable if and only if f is ∆ and ∇ differentiable. Definition 5 ([14]). Let f (·) : T → R, we define the generalized diamond exponential function bÿ Theorem 8. The generalized diamond exponential function is the solution of the diamond α-dynamic equation as follows: hence we have the solution of (9) and (10) with the initial value f (t 0 ) = 1 is equivalent to the solution of (7). On the other hand, by (5), we can obtain Therefore the dynamic Equation (5) can be reduced to i.e., f (t) with f (t 0 ) = 1 is a solution of (7). The proof is complete. (7), let T = Zh, h > 0, by Theorem 9, then the solution of (7) can be given by

Proof.
Since by Lemma 3 and Lemma 6, we can obtain The proof is complete.

t)A(t)A ρ(t) I + ν(t)A ρ(t)
On the other hand, for A(t) = e −A σ (t, t 0 ), we can obtain hence the dynamic Equation (11) can turn into: If the diamond α-dynamic Equation (11) can be given as: for any 0 ≤ α ≤ 1. Therefore thus A(t) = e −A σ (t, t 0 ). The proof is complete. , t < t 0 .
In fact, by A(t) = e −A σ (t, t 0 ), we can obtain that is For t ∈ T, t < t 0 , we can obtain for any 0 ≤ k ≤ t−t 0 h + 1, t 0 + kh > t. Therefore

Example 10.
For (12), let T = q Z , q > 1, by Theorem 11, then the solution of (12) can be given by