#### Abstract

The purpose of this paper is to develop a method for the construction of solutions to initial problems of linear discrete systems with constant coefficients and with two delays where , are fixed, , , are constant matrices, is a given vector, and is an unknown vector. Solutions are expressed with the aid of a special function called the discrete matrix delayed exponential for two delays. Such approach results in a possibility to express an initial Cauchy problem in a closed form. Examples are shown illustrating the results obtained.

#### 1. Introduction

Throughout the paper, we will use the following notation. For integers , , , we define the set . Similarly, we define the sets and . The function used below is the floor integer function. We will employ the following property of the floor integer function: where .

In this paper, we deal with the discrete system where , , are fixed, , , are constant matrices, is a given vector, and is an unknown vector.

Together with (2), we consider an initial (Cauchy) problem

Define binomial coefficients as customary; that is, for and , We recall that, for a well-defined discrete function , the forward difference operator is defined as . In the paper, we also adopt the customary notation if . In the case of double sums, we set if at least one of the inequalities , holds.

In [1, 2], a discrete matrix delayed exponential for a single delay was defined.

*Definition 1. *For an constant matrix , , and fixed , one defines the discrete matrix delayed exponential as follows:
where is an null matrix and is an unit matrix.

Such discrete matrix delayed exponential was used in [1] to construct solutions of the initial problems (2), (3) with , where is an zero matrix. In these constructions, the main property (Theorem 2) of discrete matrix delayed exponential for a single delay was utilized in [1].

Theorem 2. *Let be a constant matrix. Then, for ,
*

The properties of delayed matrix exponential functions for their continuous and discrete variants and their applications are the topic of recent papers [1–18]. We note that the definition of the delayed matrix exponential was first defined for the continuous case in [4] and, for the discrete case, in [1, 2].

The paper is organized as follows. Discrete matrix delayed exponentials for two delays and their main property are considered in Section 2. A representation of the solution to problem (2), (3) is given in Section 3 and examples illustrating the results obtained are shown in Section 4.

#### 2. Discrete Matrix Delayed Exponential for Two Delays and Its Main Property

In order to extend the results proved in [1, 2] to problems (2), (3), a discrete matrix delayed exponential for two delays was proposed in [3]. There is a discrete matrix delayed exponential for two delays , , defined as follows.

*Definition 3. *Let , be constant matrices with and let , , be fixed integers. One defines a discrete matrix function called the discrete matrix delayed exponential for two delays , and for two constant matrices , :
where

Let us show an example illustrating this special exponential function.

*Example 4. *For we will construct the matrix if and . Computing particular matrices generating for , we get
The main property of was proved in [3].

Theorem 5. *Let , be constant matrices with and let , , be fixed integers. Then
**
for .*

The analysis of applicability to a representation of the solution to initial problem (2), (3) unfortunately does not lead to satisfactory results because, as we will see below, an additional condition is necessary. A small difference in the definition results in representations of solutions of initial problems without this assumption. Now we give a second definition of a discrete matrix delayed exponential for two delays .

*Definition 6. *Let , be constant matrices with and let , , be fixed integers. One defines a discrete matrix function called the discrete matrix delayed exponential for two delays , and for two constant matrices , as follows:
where

*Remark 7. *For , it is easy to deduce that .

In order to compare both types of discrete delayed matrices for two delays and see the difference between both definitions, we consider the following example where delays are the same as in Example 4.

*Example 8. *For we will construct the matrix if and . Computing particular matrices generating for , we get

The main property of is given by the following theorem.

Theorem 9. *Let , be constant matrices with and let , , be fixed integers. Then
**
for .*

*Proof. *Let . From (1) and (13), we can see easily that, for an integer satisfying
the equation
holds by Definition 6 of . Since , we have

By the definition of the forward difference, that is,
we conclude that it is reasonable to divide the proof into four parts given by the four values of integer .

In the first case, is such that
in the second case
in the third case
and in the fourth case
We see that the above four cases cover all the possible relations between , , and .

In the proof, we use obvious identities
where and
where , derived from (4) and (24).

Now we consider (in parts (I)–(IV) below) all four cases and perform auxiliary computations. The proof will be finished in part (V).*(I)* *.* From (1) and (13), we get

Therefore, and, by Definition 6,

Similarly, omitting details, we get, using (1) and (13), and

Let . We show that

By (1),
or
From the last inequalities, we get
and (29) holds by (4). For that reason and since , we can replace by in (27). Thus, we have

It is easy to see that, due to (5), formula (33) can be used instead of (27) if too. Let . Similarly, we can show that
and, since , we can replace by in (28). Thus, we have
It is easy to see that, due to (5), formula (35) can be used instead of (28) if too By Definition 6,

Due to (1), we also conclude that
because
The second formula can be proved similarly.

Then,

Now we are able to prove that
*(II)* *.* In this case,
and . In addition to this (see the relevant computations performed in case (I)), we have and .

Then,

For , , and , we have
and, for , , and , we have
Thus, we can substitute for in (42) and for in (43).

Like with the computations performed in the previous part of the proof, (29), (34) hold. So we can substitute for in (43) and for in (44).

Accordingly, we have

It is easy to see that, due to (5), formula (48) can also be used instead of (43) if and formula (49) can also be used instead of (44) if . Therefore, we see that (like in part (I)) the relation (40) must be proved.*(III)* *.* In this case, we have (see the relevant computations in cases (I) and (II))
Then,
For , , and , we have
and, for , , and , we get
Thus we can replace by in (51) and by in (53).

Like with the computations performed in cases (I) and (II), formulas (29), (34) hold and we can substitute for in (52) and for in (53). This means that
It is easy to see that, due to (5), formula (57) can also be used instead of (52) if and formula (58) can also be used instead of (53) if . Therefore, we see that (as in parts (I), (II)) (40) must be proved.*(IV)* *.* In this case, we have (see similar combinations in the cases (II) and (III))
Then,
As in part (II), for , , and , formulas (45) hold and, for , , and , formulas (46) hold. Thus we can substitute for in (60) and for in (61).

As in part (III), for , , and , formulas (54) hold and, for , , and , formulas (55) hold. Thus we can replace by in (60) and by in (62).

As before, (29), (34) hold and we can substitute for in (61) and for in (62). Thus, we have
It is easy to see that, due to (5), formula (64) can also be used instead of (61) if and formula (65) can also be used instead of (62) if . Therefore, we see that (as in all the previous parts) (40) must be proved.*(V) The Proof of Formula (40).* Now we prove (40). With the aid of (18), (19), (24), and (36), we get
By (25), we have