Volume 2014, Issue 1 320476

Research Article

Open Access

Representation of the Solutions of Linear Discrete Systems with Constant Coefficients and Two Delays

Corresponding Author

Josef Diblík

[email protected]

Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic vutbr.cz

Department of Mathematics, Faculty of Electrical Engineering, Brno University of Technology, 616 00 Brno, Czech Republic vutbr.cz

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Blanka Morávková,

Blanka Morávková

Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic vutbr.cz

Department of Mathematics, Faculty of Electrical Engineering, Brno University of Technology, 616 00 Brno, Czech Republic vutbr.cz

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Josef Diblík,

Corresponding Author

Josef Diblík

[email protected]

Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic vutbr.cz

Department of Mathematics, Faculty of Electrical Engineering, Brno University of Technology, 616 00 Brno, Czech Republic vutbr.cz

Search for more papers by this author

Blanka Morávková,

Blanka Morávková

Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic vutbr.cz

Department of Mathematics, Faculty of Electrical Engineering, Brno University of Technology, 616 00 Brno, Czech Republic vutbr.cz

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First published: 16 April 2014

https://doi.org/10.1155/2014/320476

Citations: 14

Academic Editor: Miroslava Růžičková

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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 Δx(k) = Bx(k − m) + Cx(k − n) + f(k), where m, n ∈ ℕ, m ≠ n, are fixed, k = 0, …, ∞, B = (b_ij), C = (c_ij) are constant r × r matrices, f is a given r × 1 vector, and x is an r × 1 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 s, t, s ≤ t, 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:

(1)

where x ∈ ℝ.

In this paper, we deal with the discrete system

(2)

where m, n ∈ ℕ, m ≠ n, are fixed,

, B = (b_ij), C = (c_ij) are constant r × r matrices,

is a given r × 1 vector, and

is an r × 1 unknown vector.

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

(3)

Define binomial coefficients as customary; that is, for n ∈ ℤ and k ∈ ℤ,

(4)

We recall that, for a well-defined discrete function ω(k), the forward difference operator Δ is defined as Δω(k) = ω(k + 1) − ω(k). In the paper, we also adopt the customary notation

if i₂ < i₁. In the case of double sums, we set

(5)

if at least one of the inequalities i₂ < i₁, j₂ < j₁ holds.

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

Definition 1. For an r × r constant matrix B, k ∈ ℤ, and fixed m ∈ ℕ, one defines the discrete matrix delayed exponential as follows:

(6)

where Θ is an r × r null matrix and I is an r × r unit matrix.

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

Theorem 2. Let B be a constant r × r matrix. Then, for ,

(7)

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 m, n ∈ ℕ, m ≠ n, defined as follows.

Definition 3. Let B, C be constant r × r matrices with BC = CB and let m, n ∈ ℕ, m ≠ n, be fixed integers. One defines a discrete r × r matrix function called the discrete matrix delayed exponential for two delays m, n and for two r × r constant matrices B, C:

(8)

where

(9)

Let us show an example illustrating this special exponential function.

Example 4. For we will construct the matrix if m = 2 and n = 3. Computing particular matrices generating for , we get

(10)

The main property of

was proved in [3].

Theorem 5. Let B, C be constant r × r matrices with BC = CB and let m, n ∈ ℕ, m ≠ n, be fixed integers. Then

(11)

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 det⁡(B + C) ≠ 0 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 B, C be constant r × r matrices with BC = CB and let m, n ∈ ℕ, m < n, be fixed integers. One defines a discrete r × r matrix function called the discrete matrix delayed exponential for two delays m, n and for two r × r constant matrices B, C as follows:

(12)

where

(13)

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 m = 2 and n = 3. Computing particular matrices generating for , we get

(14)

The main property of is given by the following theorem.

Theorem 9. Let B, C be constant r × r matrices with BC = CB and let m, n ∈ ℕ, m < n, be fixed integers. Then

(15)

for

Proof. Let k ≥ 1. From (1) and (13), we can see easily that, for an integer k ≥ 0 satisfying

(16)

the equation

(17)

holds by Definition 6 of

. Since ΔI = Θ, we have

(18)

By the definition of the forward difference, that is,

(19)

we conclude that it is reasonable to divide the proof into four parts given by the four values of integer k.

In the first case, k is such that

(20)

in the second case

(21)

in the third case

(22)

and in the fourth case

(23)

We see that the above four cases cover all the possible relations between k, p_(k), and q_(k).

In the proof, we use obvious identities

(24)

where n, k ∈ ℕ and

(25)

where i, j ∈ ℕ, 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) (p_(k) − 1)(m + 1) + 1 ≤ k < p_(k)(m + 1)∧(q_(k) − 1)(n + 1) + 1 ≤ k < q_(k)(n + 1). From (1) and (13), we get

(26)

Therefore, p_(k−m) = p_(k) − 1 and, by Definition 6,

(27)

Similarly, omitting details, we get, using (1) and (13), q_(k−n) = q_(k) − 1 and

(28)

Let q_(k−m) ≥ 1. We show that

(29)

By (1),

(30)

(31)

From the last inequalities, we get

(32)

and (29) holds by (4). For that reason and since q_(k−m) ≤ q_(k), we can replace q_(k−m) by q_(k) in (27). Thus, we have

(33)

It is easy to see that, due to (5), formula (33) can be used instead of (27) if q_(k−m) < 1 too. Let p_(k−n) ≥ 1. Similarly, we can show that

(34)

and, since p_(k−n) ≤ p_(k), we can replace p_(k−n) by p_(k) in (28). Thus, we have

(35)

It is easy to see that, due to (5), formula (35) can be used instead of (28) if p_(k−n) < 1 too By Definition 6,

(36)

Due to (1), we also conclude that

(37)

because

(38)

The second formula can be proved similarly.

Then,

(39)

Now we are able to prove that

(40)

(II) k = p_(k)(m + 1)∧(q_(k) − 1)(n + 1) + 1 ≤ k < q_(k)(n + 1). In this case,

(41)

and p_(k+1) = p_(k) + 1. In addition to this (see the relevant computations performed in case (I)), we have q_(k−n) = q_(k) − 1 and q_(k+1) = q_(k).

Then,

(42)

(43)

(44)

For k = p_(k)(m + 1), i = p_(k), and j ≥ 0, we have

(45)

and, for k = p_(k)(m + 1), i = p_(k) − 1, and j ≥ 0, we have

(46)

Thus, we can substitute p_(k) − 1 for p_(k) in (42) and p_(k) − 2 for p_(k) − 1 in (43).

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

Accordingly, we have

(47)

(48)

(49)

It is easy to see that, due to (5), formula (48) can also be used instead of (43) if q_(k−m) < 1 and formula (49) can also be used instead of (44) if p_(k−n) < 1. Therefore, we see that (like in part (I)) the relation (40) must be proved.

(III) (p_(k) − 1)(m + 1) + 1 ≤ k < p_(k)(m + 1)∧k = q_(k)(n + 1). In this case, we have (see the relevant computations in cases (I) and (II))

(50)

Then,

(51)

(52)

(53)

For k = q_(k)(n + 1), j = q_(k), and i ≥ 0, we have

(54)

and, for k = q_(k)(m + 1), j = q_(k) − 1, and i ≥ 0, we get

(55)

Thus we can replace q_(k) by q_(k) − 1 in (51) and q_(k) − 1 by q_(k) − 2 in (53).

Like with the computations performed in cases (I) and (II), formulas (29), (34) hold and we can substitute q_(k) for q_(k−m) in (52) and q_(k) for q_(k−n) in (53). This means that

(56)

(57)

(58)

It is easy to see that, due to (5), formula (57) can also be used instead of (52) if q_(k−m) < 1 and formula (58) can also be used instead of (53) if p_(k−n) < 1. Therefore, we see that (as in parts (I), (II)) (40) must be proved.

(IV) k = p_(k)(m + 1)∧k = q_(k)(n + 1). In this case, we have (see similar combinations in the cases (II) and (III))

(59)

Then,

(60)

(61)

(62)

As in part (II), for k = p_(k)(m + 1), i = p_(k), and j ≥ 0, formulas (45) hold and, for k = p_(k)(m + 1), i = p_(k) − 1, and j ≥ 0, formulas (46) hold. Thus we can substitute p_(k) − 1 for p_(k) in (60) and p_(k) − 2 for p_(k) − 1 in (61).

As in part (III), for k = q_(k)(n + 1), j = q_(k), and i ≥ 0, formulas (54) hold and, for k = q_(k)(m + 1), j = q_(k) − 1, and i ≥ 0, formulas (55) hold. Thus we can replace q_(k) by q_(k) − 1 in (60) and q_(k) − 1 by q_(k) − 2 in (62).

As before, (29), (34) hold and we can substitute q_(k) for q_(k−m) in (61) and p_(k) for p_(k−n) in (62). Thus, we have

(63)

(64)

(65)

It is easy to see that, due to (5), formula (64) can also be used instead of (61) if q_(k−m) < 1 and formula (65) can also be used instead of (62) if p_(k−n) < 1. 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

(66)

By (25), we have

(67)

Now, in the first and third sum, we replace the summation index i by i + 1 and, in the second and fourth sum, we replace the summation index j by j + 1. Then,

(68)

Due to (33) and (35), we conclude that formula (40) is valid.

We proved that formula (15) holds in each of the considered cases (I), (II), (III), and (IV) for k ≥ 1. If k = 0, the proof can be done directly because p₍₀₎ = q₍₀₎ = 0, p₍₁₎ = q₍₁₎ = 1,

(69)

Formula (15) holds again. Theorem 9 is proved.

3. Representing the Solution of an Initial Problem by Discrete Matrix Delayed Exponential for Two Delays

In this part, we prove the main results of the paper. With the aid of both discrete matrix delayed exponentials we give formulas for the solution of the homogeneous and nonhomogeneous initial problem (2), (3).

3.1. Representing the Solution of a Homogeneous Initial Problem

Consider the homogeneous initial problem

(70)

(71)

First we derive formulas for the solution of (70), (71) with the aid of discrete matrix delayed exponential

and then with the aid of discrete matrix delayed exponential

Theorem 10. Let B, C be constant r × r matrices such that

(72)

and let m, n ∈ ℕ, m < n, be fixed integers. Then, the solution of the initial Cauchy problem (70), (71) can be expressed in the form

(73)

where

and

(74)

Proof. We are going to find the solution of the problem (70), (71) in the form

(75)

with unknown constant vectors v_j. Due to linearity (taking into account that k varies), we have, for k ≥ 0,

(76)

Using formula (11),

(77)

Now we conclude that, for any v_j and , the equation Δx(k) = Bx(k − m) + Cx(k − n) holds. We will try to satisfy initial conditions (71). Due to (75), we have, for ,

(78)

Due to Definition 3, for . So we have

Subtracting the neighbouring equations (En−1 − En ), (En−2 − En−1),…,(E0 − E1), we get

By Definition 3, we have

(79)

and, from the foregoing equations, we get

(80)

The previous formulas can be shortened as

(81)

where

. Finally, from (En), we get

(82)

Theorem 10 is proved.

Now we express the solution of the homogeneous Cauchy problem by . In this case, the condition det⁡(B + C) ≠ 0 is not necessary.

Theorem 11. Let B, C be constant r × r matrices with BC = CB and let m, n ∈ ℕ, m < n, be fixed integers. Then the solution of the initial Cauchy problem (70), (71) can be expressed in the form

(83)

where

and

(84)

Proof. We are going to find the solution of the problem (70), (71) in the form

(85)

with unknown constant vectors w_j. Due to linearity (taking into account that k varies), we have

(86)

We use formula (15) and we get

(87)

Now we conclude that, for any w_j and , the equation Δx(k) = Bx(k − m) + Cx(k − n) holds. We will try to satisfy initial conditions (71). Due to (83), we have, for ,

(88)

By Definition 6, we have for and for . Thus, we have

We see directly that w_n = φ(−n). Subtracting the neighbouring equations (E~n-1-E~n), (E~n-2-E~n-1),…,(E~n-m-E~n-m+1), we immediately get the formulas for w_n−1, w_n−2, …, w_n−m as follows:

Further, subtracting the neighbouring equations (E~n-m-1-E~n-m), (E~n-m-2-E~n-m-1),…,(E~0-E~1), we get

The previous formulas can be written as

(89)

Theorem 11 is proved.

3.2. Representing the Solution of a Nonhomogeneous Initial Problem

We consider a nonhomogeneous initial Cauchy problem

(90)

(91)

By the theory of linear equations, we can obtain its solution as the sum of a solution of adjoint homogeneous problem (70), (71) (satisfying the same initial data) and a particular solution of (90) being zero on an initial interval. Let us, therefore, find such a particular solution.

We need an auxiliary lemma the proof of which is omitted.

Lemma 12. Let a function F(k, n) of two discrete variables be given. Then,

(92)

Now we are ready to find a particular solution x_p(k),

of the initial Cauchy problem:

(93)

(94)

Theorem 13. The solution x = x_p(k) of the initial Cauchy problem (93), (94) can be represented on in the form

(95)

Proof. We are going to find a particular solution x_p(k) of problem (93), (94) in the form (95). We substitute (95) into (93). Then, we get

(96)

We modify the left-hand side of (96). With the aid of Lemma 12, we obtain

(97)

and, applying Theorem 9, we get

(98)

By Definition 6, we have

for

and

for

. Thus, we get

(99)

and (96) holds.

Combining the results of Theorems 10, 11, and 13, we get immediately the following two theorems, which describe the solution of (90), (91). The first theorem uses the delayed matrix exponential and the second one uses the delayed matrix exponential .

Theorem 14. Let B, C be constant r × r matrices with

(100)

and let m, n ∈ ℕ, m < n, be fixed integers. Then, the solution of the initial Cauchy problem (90), (91) can be expressed in the form

(101)

where

and

(102)

Theorem 15. Let B, C be constant r × r matrices with BC = CB and let m, n ∈ ℕ, m < n, be fixed integers. Then, the solution of the initial Cauchy problem (90), (91) can be expressed in the form

(103)

where

and

(104)

4. Examples

Below, we show four examples to demonstrate the results achieved.

Example 16. Let us represent the solution of the scalar (r = 1) problem (70), (71) where we put m = 2, n = 3, B = b, C = c, φ(−3) = 1, φ(−2) = 2, φ(−1) = 3, and φ(0) = 4, using Theorem 10. We get

(105)

(106)

By Theorem 10, the solution of problem (105), (106) is

(107)

where

(108)

Thus, we get

(109)

We give values of x(k) for as follows:

(110)

Example 17. Let us represent the solution of the scalar (r = 1) problem (90), (91) where we put m = 2, n = 3, B = b, C = c, φ(−3) = 1, φ(−2) = 2, φ(−1) = 3, φ(0) = 4, and f(k) = k + 1, using Theorem 11. Thus, we have

(111)

(112)

By Theorem 11, the solution of problem (111), (112) is

(113)

where

(114)

Thus, we get

(115)

The first eight values of the homogeneous problem are given in Example 16. Now, we compute the first eight values of a particular solution

as follows:

(116)

Together, we get

(117)

Example 18. Let us represent the solution of the scalar (r = 1) problem (70), (71) where we put m = 2, n = 3, B = b = 4, C = c = −1, φ(−3) = 1, φ(−2) = 2, φ(−1) = 3, and φ(0) = 4, using Theorem 10. Thus, we have

(118)

(119)

By Theorem 10, the solution of problem (118), (119) is

(120)

where

(121)

Thus, we get

(122)

Example 19. Let us represent the solution of the scalar (r = 1) problem (90), (91) where we put m = 2, n = 3, B = b = 4, C = c = −1, φ(−3) = 1, φ(−2) = 2, φ(−1) = 3, φ(0) = 4, and f(k) = k + 1, using Theorem 11. Thus, we have

(123)

(124)

By Theorem 11, the solution of the problem (123), (124) is

(125)

where

(126)

Thus, we get

(127)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The first author was supported by the Grant 201/10/1032 of the Czech Grant Agency (Prague).

References

1 Diblík J. and Khusainov D. Ya., Representation of solutions of linear discrete systems with constant coefficients and pure delay, Advances in Difference Equations. (2006) 2006, 13, 80825, MR2238982, ZBL1139.39027.
10.1155/ADE/2006/80825
Web of Science® Google Scholar
2 Diblík J. and Khusainov D. Ya., Representation of solutions of discrete delayed system x(k + 1) = Ax(k) + Bx(k − m) + f(k) with commutative matrices, Journal of Mathematical Analysis and Applications. (2006) 318, no. 1, 63–76, https://doi.org/10.1016/j.jmaa.2005.05.021, MR2210872, ZBL1094.39002.
10.1016/j.jmaa.2005.05.021
Web of Science® Google Scholar
3 Diblík J. and Morávková B., Discrete matrix delayed exponential for two delays and its property, Advances in Difference Equations. (2013) 2013, article 139, 18, https://doi.org/10.1186/1687-1847-2013-139, MR3068650.
10.1186/1687-1847-2013-139
Google Scholar
4 Khusainov D. Ya. and Shuklin G. V., Linear autonomous time-delay system with permutation matrices solving, Studies of the University of Žilina. Mathematical Series. (2003) 17, no. 1, 101–108, MR2064983, ZBL1064.34042.
Google Scholar
5 Boichuk A., Diblík J., Khusainov D., and Růžičková M., Boundary value problems for delay differential systems, Advances in Difference Equations. (2010) 2010, 20, 593834, MR2660943, ZBL1204.34087.
10.1186/1687-1847-2010-593834
Google Scholar
6 Boichuk A., Diblík J., Khusainov D., and Růžičková M., Fredholm′s boundary-value problems for differential systems with a single delay, Nonlinear Analysis: Theory, Methods & Applications. (2010) 72, no. 5, 2251–2258, https://doi.org/10.1016/j.na.2009.10.025, MR2577792, ZBL1190.34073.
10.1016/j.na.2009.10.025
Web of Science® Google Scholar
7 Boichuk A., Diblík J., Khusainov D., and Růžičková M., Boundary-value problems for weakly nonlinear delay differential systems, Abstract and Applied Analysis. (2011) 2011, 19, 631412, https://doi.org/10.1155/2011/631412, MR2802836, ZBL1222.34075.
10.1155/2011/631412
Google Scholar
8 Diblík J., Fečkan M., and Pospíšil M., Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices, Ukrains′kyi Matĕmatyčhnyi Žhurnal. (2013) 65, 58–69.
Google Scholar
9 Diblík J., Fečkan M., and Pospíšil M., Representation of a solution of the Cauchy problem for an oscillating system with multiple delays and pairwise permutable matrices, Abstract and Applied Analysis. (2013) 2013, 10, 931493, MR3055976, ZBL06209501.
10.1155/2013/931493
Google Scholar
10 Diblík J., Khusainov D., Kukharenko O., and Svoboda Z., Solution of the first boundary-value problem for a system of autonomous second-order linear partial differential equations of parabolic type with a single delay, Abstract and Applied Analysis. (2012) 2012, 27, 219040, MR2955027, ZBL1250.35117.
10.1155/2012/219040
Google Scholar
11 Diblík J., Khusainov D. Ya., Lukáčová J., and Růžičková M., Control of oscillating systems with a single delay, Advances in Difference Equations. (2010) 2010, 15, 108218, MR2595647, ZBL1184.93015.
10.1186/1687-1847-2010-108218
Google Scholar
12 Diblík J., Khusainov D. Ya., and Růžičková M., Controllability of linear discrete systems with constant coefficients and pure delay, SIAM Journal on Control and Optimization. (2008) 47, no. 3, 1140–1149, https://doi.org/10.1137/070689085, MR2407011, ZBL1161.93004.
10.1137/070689085
Web of Science® Google Scholar
13 Khusainov D. Ya., Diblík J., Růžičková M., and Lukáčová J., Representation of a solution of the Cauchy problem for an oscillating system with pure delay, Nonlinear Oscillations. (2008) 11, no. 2, 276–285, https://doi.org/10.1007/s11072-008-0030-8, MR2510692.
10.1007/s11072-008-0030-8
Google Scholar
14 Khusainov D. Ya. and Shuklin G. V., On relative controllability in systems with pure delay, International Applied Mechanics. (2005) 41, no. 2, 118–130, https://doi.org/10.1007/s10778-005-0079-3, MR2190935.
10.1007/s10778-005-0079-3
Web of Science® Google Scholar
15 Medved′ M. and Pospíšil M., Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts defined by pairwise permutable matrices, Nonlinear Analysis: Theory, Methods & Applications. (2012) 75, no. 7, 3348–3363, https://doi.org/10.1016/j.na.2011.12.031, MR2891173, ZBL1244.34096.
10.1016/j.na.2011.12.031
Web of Science® Google Scholar
16 Medved′ M., Pospíšil M., and Škripková L., Stability and the nonexistence of blowing-up solutions of nonlinear delay systems with linear parts defined by permutable matrices, Nonlinear Analysis: Theory, Methods & Applications. (2011) 74, no. 12, 3903–3911, https://doi.org/10.1016/j.na.2011.02.026, MR2802976.
10.1016/j.na.2011.02.026
Web of Science® Google Scholar
17 Medved′ M. and Škripková L., Sufficient conditions for the exponential stability of delay difference equations with linear parts defined by permutable matrices, Electronic Journal of Qualitative Theory of Differential Equations. (2012) no. 22, 1–13, MR2900482.
10.14232/ejqtde.2012.1.22
Web of Science® Google Scholar
18 Pospíšil M., Representation and stability of solutions of systems of functional differential equations with multiple delays, Electronic Journal of Qualitative Theory of Differential Equations. (2012) no. 54, 1–30, MR2959044.
10.14232/ejqtde.2012.1.54
Web of Science® Google Scholar

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Representation of the Solutions of Linear Discrete Systems with Constant Coefficients and Two Delays

Abstract

1. Introduction

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

3. Representing the Solution of an Initial Problem by Discrete Matrix Delayed Exponential for Two Delays

3.1. Representing the Solution of a Homogeneous Initial Problem

3.2. Representing the Solution of a Nonhomogeneous Initial Problem

4. Examples

Conflict of Interests

Acknowledgment

References

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References

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Representation of the Solutions of Linear Discrete Systems with Constant Coefficients and Two Delays

Abstract

1. Introduction

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

3. Representing the Solution of an Initial Problem by Discrete Matrix Delayed Exponential for Two Delays

3.1. Representing the Solution of a Homogeneous Initial Problem

3.2. Representing the Solution of a Nonhomogeneous Initial Problem

4. Examples

Conflict of Interests

Acknowledgment

References

Citing Literature

References

Related

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