General solutions of weakly delayed discrete systems in 3D
-
Josef Diblík
,
Hana Boháčková
Abstract
Discrete systems
1 Introduction
The article deals with three-dimensional discrete systems of weakly delayed equations. Weakly delayed systems form a class of systems such that their solutions have a special “merging” property, meaning that some solutions, being defined by different initial data, continue, after an interval, as a single solution. In the case considered, the set of all initial data splits into three mutually different subsets with the initial data of each subset generating solutions that merge into a single one after passing an interval. Then, it is possible to find simple formulas for general solutions, which is the aim of the present article.
Weakly delayed discrete planar systems are investigated in [15], where, using
1.1 Weakly delayed systems
Throughout the article, the following notation is used:
In this article, we investigate discrete systems with constant coefficients and with a single delay
where
The initial problem (1), (2) defines a unique solution of (1) on
The characteristic equation to (1),
is derived in the usual way if we look for a solution to (1) in the form
Definition 1
System (1) is called weakly delayed if, for every
In the sequel, rather than repeating the term weakly delayed system, we will use the abbreviation WD-system, etc. An important property formulated by the below lemma enables regular transformations of WD-systems.
Lemma 1
If the system (1) is a WD-system, then its arbitrary transformation
1.2 Problem under consideration
The article treats WD-systems (1) solving the problem of finding sharp formulas for the general solutions that solely depend on three parameters when a transient interval has been passed. The problem is solved completely, for every possible Jordan form of the matrix
1.3 Structure of the article
After the general solutions are constructed for all possible WD-systems (1), Section 1.4 introduces some preliminaries. Here, the general coefficient condition for a system (1) to be a WD-system is given with some properties of the matrix
1.4 Coefficient criterion for WD-systems and properties of matrix
B
Here, we formulate a general coefficient criterion indicating when system (1) is a WD-system. It directly follows from a detailed analysis of equation (4) (we refer to [34]).
Theorem 1
System (1) is a WD-system if and only if
The following properties are obvious. If (5), (6), and (8) hold, then all eigenvalues of matrix
2 Specific criteria for WD-systems
In this section, we formulate specific criteria for WD-systems derived from Theorem 1. These will depend on the Jordan forms of the matrix
Recall that all eigenvalues of
that can be obtained from (1) by transformation
are, by Lemma 1, again WD-systems. The initial data for (12) derived from (2) are
Remark 1
Now, in the sequel, without loss of generality, we can always assume that the matrix
where
2.1 Jordan canonical forms of
A
and their powers
In this section, we recall all possible Jordan forms of matrix
If (11) has real distinct roots
If (11) has two distinct roots – a single root
if
if
if
if
if
It is easy to see that
where
If the complex conjugate roots are given in the exponential form
and
2.2 Specific criteria
The following formulated criteria are consequences of conditions (5)–(10) in Theorem 1. Their proofs can be done by computing the relevant determinants and analyzing the arising expressions.
2.2.1 Criterion for WD-systems in case (15)
Consider system (1) with the matrix
Theorem 2
System (23) is a WD-system if and only if
2.2.2 Criterion for WD-systems in case (16)
Consider system (1) with the matrix
Theorem 3
System (28) is a WD-system if and only if
2.2.3 Criterion for WD-systems in case (17)
Consider system (1) with the matrix
Theorem 4
System (34) is a WD-system if and only if
2.2.4 Criterion for WD-systems in case (18)
Consider system (1) with the matrix
Theorem 5
System (41) is a WD-system if and only if
2.2.5 Criterion for WD-systems in case (19)
Consider system (1) with the matrix
Theorem 6
System (45) is a WD-system if and only if
2.3 Criterion for WD-systems in case (20)
Consider system (1) with the matrix
Theorem 7
System (51) is a WD-system if and only if
2.3.1 Criterion for WD-systems in case (21)
Consider system (1) with the matrix
Theorem 8
System (58) is a WD-system if and only if
3 Associated nondelayed system
In this section, we will transform system (1) into a higher dimensional nondelayed system to find its general solution. Define new dependent functions
where
where
is a
Let us transform system (66) using a suitable transformation
where
where
The initial data for system (70) are, as it follows from (68), (69),
The solution of initial problems (70) and (72) is given by formula
and the solution of initial problems (66) and (68) is
If the matrix
3.1 Properties of matrices used
Before studying the possible Jordan forms of the matrix
3.1.1 Geometric multiplicities of identical eigenvalues of
A
and
G
Recall some obvious facts. Eigenvalues of matrices
3.1.2 Relationship between the eigenvalues of matrices
A
,
B
, and
A
Theorem 9
Let (1) be a WD system. Then, the set of all the eigenvalues of the matrix
Proof
Let
where
Consider equation
As it follows from (76), equation (78) has exactly
and nonzero roots satisfy
As a consequence of the conclusion of Theorem 9, one can expect that the respective geometric multiplicities of eigenvalues of the matrices
Example 1
Consider weakly delayed system (41) with
Then,
and
Nevertheless, contrary to the property described in Example 1, the following theorem can be proved.
Theorem 10
The geometric multiplicity of the zero eigenvalue of the matrix
Proof
Let us find the geometric multiplicity of the zero eigenvalue of the matrix
where
Now it is clear that the number of linearly independent eigenvectors of matrix
4 Jordan forms of the associated-system matrix
When using formula (73), it is necessary to compute the powers of the matrix
The results of the computations depend on the geometric multiplicity of the zero eigenvalue of matrix
Next we give formulas for powers of the relevant matrix
and
Similarly, instead of
4.1 Jordan forms of matrices by Weyr’s algorithm
For each high-dimensional matrix in Sections 4.3–4.9, its Jordan form will be established using Weyr’s algorithm [4,11,23,35]. We will use only the part of this algorithm described below (following the explanation given in [23]). Let
their ranks
and nullities
where the number
we have
associated with the eigenvalue
The system of eigenvectors and generalized eigenvectors to an eigenvalue of
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
4.2 Auxiliary matrices and their powers
Below we use some auxiliary matrices. Define an
and a frequently used in the article matrix
In particular,
4.3 The form and powers of
G
if
A
=
A
1
4.3.1 The case of
m
g
(
B
,
0
)
=
1
In this case, there exists a transition matrix
Then,
Then
4.3.2 The case of
m
g
(
B
,
0
)
=
2
In this case, there exists a transition matrix
and, as the formula for determining the rank
we obtain
To compute the next rank
Let us begin with the determination of an auxiliary rank
We perform the following operations on rows, where each number in round brackets refers to the number of a row in the last matrix:
Since
Therefore,
The characteristic numbers are
and, therefore, Table 1 will have two columns (with the first one consisting of
Reduction of Table 1 if
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Because the roots
Then
4.4 The form and powers of
G
if
A
=
A
2
4.4.1 Preliminaries
Consider an auxiliary matrix
By (30)–(33),
Theorem 11
If
Proof
Let us find
where, see (75),
We obtain
Therefore, taking into account that, by (29),
that is,
4.4.2 The case of
m
g
(
B
,
0
)
=
1
Then,
Then
Then
4.4.3 The case of
m
g
(
B
,
0
)
=
2
To compute
We perform the following operations on rows, where each number in round brackets refers to the number of a row in the last matrix:
By using formula (30) and the fact that all determinants of the second order of
Therefore,
Since
Table 1 will have two columns taking the same form as in Table 3 if
Then
Then
4.5 The form and powers of
G
if
A
=
A
3
4.5.1 Preliminaries
Consider an auxiliary matrix
By (39) and (40), we have
Theorem 12
If
Proof
Let us find
where, see (75),
We obtain
Therefore, taking into account that, by (35),
that is,
4.5.2 The case of
m
g
(
B
,
0
)
=
1
Then,
Then
Then, the associated Jordan form of
Then
4.5.3 The case of
m
g
(
B
,
0
)
=
2
By using formulas (36) and the property that all determinants of the second order of
We perform the following operations on the rows, where each number in round brackets refers to the number of a row in the last matrix:
and then
Now we continue as in Section 4.4.3. Reduction of Table 1 if
Then
Then
4.6 The form and powers of
G
if
A
=
A
4
4.6.1 Preliminaries
Here, we explain why, in this case, the matrix of type
Theorem 13
If
Proof
Let us find
where, see (75),
Immediately from (99) and (100), we obtain
and
In the previous sections, the matrix of linear terms of the system similar to (104)–(106) was denoted by
Now it is clear that, if
4.6.2 The case of
m
g
(
B
,
0
)
=
1
In this case, by Theorem 13, we have
Then,
Then
4.6.3 The case of
m
g
(
B
,
0
)
=
2
In the case considered, by Theorem 13, we have
To compute
We perform the following operations on rows, where each number in round brackets refers to the number of a row in the last matrix:
By using formula (42) and the property
Therefore,
Since
Table 1 will have only two columns and takes the same form as in Table 3 if
Then
4.7 The form and powers of
G
if
A
=
A
5
4.7.1 Preliminaries
Define an auxiliary matrix
Obviously,
Theorem 14
If
Proof
Let us find
where, see (75),
Immediately from (107), we obtain
Therefore, taking into account that, by (47),
that is,
4.7.2 The case of
m
g
(
B
,
0
)
=
1
Then,
Then
Then
Then
4.7.3 The case of
m
g
(
B
,
0
)
=
2
To compute
We perform the following operations on rows, where each number in round brackets refers to the number of a row in the last matrix:
By using the formula (46) and the property
Therefore,
Since
Table 1 (where
Then
Then
Then
4.8 The form and powers of
G
if
A
=
A
6
4.8.1 Preliminaries
Define an auxiliary matrix
Obviously,
Theorem 15
If
Proof
Let us find
where, see (75),
Immediately from (115) and (116), we obtain
Therefore, taking into account that, by (54),
that is,
4.8.2 The case of
m
g
(
B
,
0
)
=
1
Then,
Then
Then
Then
4.8.3 The case of
m
g
(
B
,
0
)
=
2
To compute
We perform the following operations on rows, where each number in round brackets refers to the number of a row in the last matrix:
Using formulas (52) and (53) and the property
Therefore,
Since
Table 1 will only have two columns taking the same form as in Table 3 (where
Then
Then
Then
4.9 The form and powers of
G
if
A
=
A
7
4.9.1 The case of
m
g
(
B
,
0
)
=
1
In this case, we can proceed in much the same way as in Section 4.3. We conclude that there exists a transition matrix
Then
4.9.2 The case of
m
g
(
B
,
0
)
=
2
In this case, we can proceed in much the same way as in Section 4.3.2. By Section 3.1.1 and Theorem 10, we have
To compute
We perform the following operations among rows, where each number in round brackets refers to the number of a row in the last matrix:
where
Therefore,
Since
Table 1 takes the form of Table 3 (constructed for
Then
5 Formulas for the general solution of problem (1), (2)
The initial data for system (1) are defined by (2). The solution of the initial problem (66), (68), that is, the problem
is (we refer to formulas (69)–(74))
where the relation between
Using an auxiliary matrix
formula (65) (i.e.,
where the initial data
Theorem 16
Let the Jordan form of A be
where
If
Theorem 17
Let the Jordan form of A be
where
if
if
if
Theorem 18
Let the Jordan form of A be
where
if
if
if
Theorem 19
Let the Jordan form of A be
where
If
Theorem 20
Let the Jordan form of A be
where
if
if
if
if
if
Theorem 21
Let the Jordan form of A be
where
if
if
if
if
if
Theorem 22
Let the Jordan form of A be
where
If
6 Initial values defining the merging of solutions
If
More exactly, if
Similarly, if
In the following theorem, formulas for computing the above-mentioned three initial values (158) through initial values
Define vectors
Theorem 23
Initial data (158) are defined by formulas
where
and
Proof
The initial data (158) are defined as suitable linear combinations of
Multiplying (160) successively by vectors
From Theorem 23, formula (159), we conclude that there exist linear combinations involving initial values
Let real numbers
where
where
where
Theorem 24
7 Examples
The results of the article will be now illustrated by three examples. In the first one, computations are performed in detail while in the remaining ones, parts of computations are omitted.
Example 2
Let system (1) be reduced to system (23), where
that is, to the system
Conditions (24)–(27) are fulfilled, system (164)–(166) is a WD-system and
and
The general solution (164)–(166) is expressed by the formula:
For
The general solution (167) depends only on parameters
must be identical for
or if
we obtain, by (168)–(170), the same values
We conclude that the solutions of system (164)–(166) defined by different initial data (171) and (172) are identical for
Example 3
Let system (1) be reduced to system (41), where
that is, to the system
Conditions (42)–(44) are fulfilled, system (173)–(175) is a WD-system and
and
Then, the general solution of the system (173)–(175) is expressed by the formula
where
The aforementioned general solution of system (173)–(175) depends on only three initial data
Now, we will demonstrate (in accordance with Theorem 24, case
or
give the same values
and the solutions of system (173)–(175) defined by (176) and (177) are identical for
Example 4
Let system (1) be reduced to system (51), where
that is, to the system
Conditions (52)–(57) are fulfilled, system (178)–(180) is a WD-system and
and
and
Then, the general solution of the system (178)–(180) is expressed by the formula
where
The aforementioned general solution of system (178)–(180) depends on only three initial data
Now, we will demonstrate (in accordance with Theorem 24, case
and
give the same new initial values
Initial data (181) and (182) define two solutions of the system (178)–(180) that are identical for
8 Remarks, conclusions, and open problems
Since Section 2, Remark 1, we have assumed, without loss of generality, that the matrix
defined by initial data (14), that is,
then can be transformed into solutions of system (1) by the substitution (13), that is, by the formula
where (see (124))
and the initial data
The article considers WD-systems (1) and their explicit general solutions are found by transforming them into nondelayed discrete higher dimensional systems. Simple formulas are found that describe general solutions depending on three arbitrary parameters only (rather than on the given
The property of the geometric multiplicities described in Section 3.1, the properties of auxiliary matrices
Discrete planar WD-systems are investigated in [15,16,20]. The criteria formulated in Theorems 2–7 (i.e., the cases of
Papers [7–10] consider mathematical models of endocrine regulation applying the so-called finite-dimension (FD) reducible differential delayed systems
Let us compare FD-reducible systems and WD-systems. FD-reducible systems are characterized by there existing a constant matrix
It describes a linearized model of the Mach number dynamics concerning the time-optimal control of a high-speed closed-circuit wind tunnel. In system (183)–(185),
In [15,16], a concept of conditional stability for planar WD-systems is formulated with relevant results proved. Generalizing this concept to three-dimensional discrete WD-systems may be a challenge for further research. Another problem is to construct nondelayed discrete systems having, on an interval, general solutions coinciding with general solutions of WD-systems. For a planar system, such results can be found in [15]. In addition, as our approach gives exact results, it might be usefull to write a computer program detecting WD-systems and producing explicit forms of general solutions. Another important challenge is to consider WD-systems of arbitrary dimension, developing the coefficient criteria and giving exact formulas for their solutions. Also, in the future, it might be useful to apply the known concrete formulas to representing the solutions of delayed systems of discrete equations, such as those derived and used, for instance, in papers [21,25,28,29,32], to obtain concrete formulas representing general solutions for higher dimensional discrete delayed systems. Attention deserves a generalization of the results for the case of the independent variable changing over a time-scale (for time-scales theory we refer to [3]) and as well as an analysis of connections between the solutions of WD-systems and the discrete variants of systems obtained by modifying the Euler-Poincaré equations [2]. Let us briefly comment on the recent papers [13,17,18]. In [13], the asymptotic behavior of solutions of fractional discrete equations is studied by the retract apparatus while, in [17], this technique is applied to studying the asymptotic behavior of solutions of Emden-Fowler second-order discrete equations. An original geometrical method is developed in [18] to investigate the behavior of solutions of complex differential equations in a neighborhood of a singular point. All the results derived in the aforementioned three articles have qualitative character and do not give exact formulas describing solutions as the present paper does. Nevertheless, it is an open question whether, for some particular cases of fractional discrete equations considered in [13], it will be possible to develop a WD-system classification and derive exact formulas for their solutions. Finally, for rudiments of theory of discrete equations, we refer to [1,5,19,24,30,33].
Acknowledgments
The authors greatly appreciate the work of the anonymous referees and the Editor, whose comments and suggestions have helped to improve this article in many aspects.
-
Funding information: The research of M. Růžičková has been supported by the Polish Ministry of Science and Higher Education under a subsidy for maintaining the research potential of the Faculty of Mathematics, University of Białystok, Poland.
-
Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.
-
Conflict of interest: The authors state no conflict of interest.
-
Data availability statement: Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
References
[1] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, CRC Press, Boca Raton, Florida, United States, 2000, DOI: http://doi.org/10.1201/9781420027020.Search in Google Scholar
[2] A. Bloch, M. F. Puiggalí, and D. M. de Diego, Metriplectic Euler-Poincaré equations: smooth and discrete dynamics, Commun. Anal. Mech. 16 (2024), no. 4, 910–927, https://doi.org/10.3934/cam.2024040. Search in Google Scholar
[3] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birghäuser, Basel, Switzerland, 2001, DOI: https://doi.org/10.1007/978-1-4612-0201-1.Search in Google Scholar
[4] O. Borůvka, Poznámka o použití Weyrovy theorie matic k integraci systému diferenciálních rovnic s konstantními koeficienty (Remark on the use of Weyras theory of matrices for the integration of systems of linear differential equations with constant coefficients), Časopis Pěst. Mat. 79 (1954), no. 2, 151–155. 10.21136/CPM.1954.117115Search in Google Scholar
[5] S. N. Elaydi, An Introduction to Difference Equations, Third Edition, Springer, New York City, USA, 2005. Search in Google Scholar
[6] E. N. Chukwu, Stability and Time-Optimal Control of Hereditary Systems. With Application to the Economic Dynamics of the U.S. Second edition, Series on Advances in Mathematics for Applied Sciences, vol. 60, World Scientific Publishing Co., Inc., River Edge, NJ, 2001, https://doi.org/10.1142/4745. Search in Google Scholar
[7] A. Churilov and A. Medvedev, An impulse-to-impulse discrete-time mapping for a time-delay impulsive system, Automatica J. IFAC 50 (2014), no. 8, 2187–2190, https://doi.org/10.1016/j.automatica.2014.05.021. Search in Google Scholar
[8] A. Churilov, A. Medvedev, and P. Mattsson, Analysis of a pulse-modulated model of endocrine regulation with time-delay, IEEE Xplore: 04 February 2013, 51st IEEE Conference on Decision and Control, 2012, Maui, pp. 362–367, https://doi.org/10.1109/CDC.2012.6426771. Search in Google Scholar
[9] A. Churilov, A. Medvedev, and P. Mattsson, Finite-dimensional reducibility of time-delay systems under pulse-modulated feedback, IEEE Xplore: 10 March 2014, 52nd IEEE Conference on Decision and Control, 2013, Firenze, pp. 2078–2083, https://doi.org/10.1109/CDC.2013.6760188. Search in Google Scholar
[10] A. Churilov, A. Medvedev, and P. Mattsson, Periodical solutions in a pulse-modulated model of endocrine regulation with time-delay, IEEE Trans. Automat. Control 59 (2014), no. 3, 728–733, https://doi.org/10.1109/TAC.2013.2273284. Search in Google Scholar
[11] J. Čermák, On a new method of solving homogeneous systems of linear difference equations with constant coefficients, Ann. Polon. Math. 1 (1954), 195–202. 10.4064/ap-1-1-195-202Search in Google Scholar
[12] A. Derevianko and J. Diblík, Planar linear differential weakly delayed systems with constant matrices and equivalent ordinary differential systems, J. Math. Anal. Appl. 552 (2025), no. 1, Paper No. 129740, 29 pp., https://doi.org/10.1016/j.jmaa.2025.129740. Search in Google Scholar
[13] J. Diblík, Bounded solutions to systems of fractional discrete equations, Adv. Nonlinear Anal. 11 (2022), no. 1, 1614–1630, https://doi.org/10.1515/anona-2022-0260. Search in Google Scholar
[14] J. Diblík, H. Halfarová, and J. Safarík, Discrete analogue of a linear differential model with delay of the Mach number dynamics and its solution, Conference Proceedings of the 9th International Conference on Mathematics and Computers in Sciences and Industry (MCSI), IEEE Computer Society, 2024, pp. 174–179. https://doi.ieeecomputersociety.org/10.1109/MCSI63438.2024.00036. Search in Google Scholar
[15] J. Diblík, H. Halfarová, and J. Šafařík, Formulas for the general solution of weakly delayed planar linear discrete systems with constant coefficients and their analysis, Appl. Math. Comput. 358 (2019), 363–381, https://doi.org/10.1016/j.amc.2019.03.068. Search in Google Scholar
[16] J. Diblík, H. Halfarová, and J. Šafařík, Two-parameters formulas for general solution to planar weakly delayed linear discrete systems with multiple delays, equivalent non-delayed systems, and conditional stability, Appl. Math. Comput. 459 (2023), Paper No. 128270, 14 pp., https://doi.org/10.1016/j.amc.2023.128270. Search in Google Scholar
[17] J. Diblík and E. Korobko, Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type, Adv. Nonlinear Anal. 12 (2023), no. 1, Paper No. 20230105, 23 pp., https://doi.org/10.1515/anona-2023-0105. Search in Google Scholar
[18] J. Diblík and M. Růžičková, Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point, Adv. Nonlinear Anal. 13 (2024), no. 1, Paper No. 20230120, 44 pp., 10.1515/anona-2023-0120. Search in Google Scholar
[19] M. S. Fadali and A. Visioli, Digital Control Engineering, Analysis and Design, Third Edition, Academic Press in an imprint of Elsevier, Elsevier, 2019. Search in Google Scholar
[20] H. Halfarová (H. Boháčková), Weakly Delayed Linear Planar Systems of Discrete Equations, Doctoral Thesis, Brno University of Technology, Brno, Czech Republic, 2014. Search in Google Scholar
[21] X. Jin, J. Wang, and D. Shen, Representation and stability of solutions for impulsive discrete delay systems with linear parts defined by non-permutable matrices, Qual. Theory. Dyn. Syst. 21 (2022), no. 4, Paper No. 152, 34 pp., https://doi.org/10.1007/s12346-022-00685-9. Search in Google Scholar
[22] A. V. Kim and A. V. Ivanov, Systems with Delays: Analysis, Control, and Computations, John Wiley & Sons, Inc.. Hoboken, NJ; Scrivener Publishing, Salem, MA, 2015, 10.1002/9781119117841. Search in Google Scholar
[23] J. Klimek, An algorithm for the construction of the fundamental system of solutions for the linear discrete systems with constant coefficients, Tatra Mt. Math. Publ. 38 (2007), 101–109. Search in Google Scholar
[24] V. Lakshmikantham and D. Trigiante, Theory of Difference Equations, Second Edition, Marcel Dekker, Inc., Madison Avenue, New York, NY, 2002, DOI: https://doi.org/10.1201/9780203910290.Search in Google Scholar
[25] C. Liang, J. R. Wang, and M. Fečkan, A study on ILC for linear discrete systems with single delay, J. Difference Equ. Appl. 24 (2018), no. 3, 358–374, https://doi.org/10.1080/10236198.2017.1409220. Search in Google Scholar
[26] A. Manitius, Feedback controllers for a wind tunnel model involving a delay: analytical design and numerical simulation, IEEE Trans. Automat. Control 29 (1984), no. 12, 1058–1068, 10.1109/TAC.1984.1103436. Search in Google Scholar
[27] A. Manitius and H. Tran, Numerical simulation of a non-linear feedback controller for a wind tunnel model involving a time delay, Opt. Control Appl. Meth. 7 (1986), 19–39, https://doi.org/10.1002/oca.4660070103. Search in Google Scholar
[28] N. I. Mahmudov, Delayed linear difference equations: the method of Z-transform, Electron. J. Qual. Theory Differ. Equ. 2020 (2020), Paper No. 53, 12 pp., 10.14232/ejqtde.2020.1.53. Search in Google Scholar
[29] N. I. Mahmudov, Representation of solutions of discrete linear delay systems with non permutable matrices, Appl. Math. Lett. 85 (2018), 8–14, https://doi.org/10.1016/j.aml.2018.05.015. Search in Google Scholar
[30] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Third Edition, Pearson, London, UK, 2009. Search in Google Scholar
[31] V. M. Popov, Pointwise degeneracy of linear, time-invariant, delay-differential equations, J. Differential Equations 11 (1972), 541–561, https://doi.org/10.1016/0022-0396(72)90066-6. Search in Google Scholar
[32] M. Pospíšil, Representation of solutions of delayed difference equations with linear parts given by pairwise permutable matrices via Z-transform, Appl. Math. Comput. 294 (2017), 180–194, https://doi.org/10.1016/j.amc.2016.09.019. Search in Google Scholar
[33] J. G. Proakis and D. G. Manolakis, Digital Signal Processing, Principles, Algorithms, and Applications, Fourth Edition, Pearson, London, UK, 2006. Search in Google Scholar
[34] J. Šafařík, Weakly Delayed Systems of Linear Discrete Equations in R3, Doctoral Thesis, Brno University of Technology, Brno, Czech Republic, 2018. Search in Google Scholar
[35] E. Weyr, O theorii forem bilinearnacute;ch, Spisy poctěné jubilejní cenou král. české spol. nauk, Prague, 1889; reprinted under the title Zur Theorie der bilinearen Formen, Monatsh. f. Mathematik und Physik 1 (1890), 163–236, https://doi.org/10.1007/BF01692474. Search in Google Scholar
© 2025 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Research Articles
- Incompressible limit for the compressible viscoelastic fluids in critical space
- Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
- Intervals of bifurcation points for semilinear elliptic problems
- On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
- Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
- Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
- Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
- Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
- Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
- Existence results for non-coercive problems
- Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
- Geometric characterization of generalized Hajłasz-Sobolev embedding domains
- Subharmonic solutions of first-order Hamiltonian systems
- Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
- Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
- Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
- Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
- Homoclinic solutions in periodic partial difference equations
- Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
- Properties of minimizers for L2-subcritical Kirchhoff energy functionals
- Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
- Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
- Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
- Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
- Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
- Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
- Existence of positive radial solutions of general quasilinear elliptic systems
- Low Mach number and non-resistive limit of magnetohydrodynamic equations with large temperature variations in general bounded domains
- Sharp viscous shock waves for relaxation model with degeneracy
- Bifurcation and multiplicity results for critical problems involving the p-Grushin operator
- Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
- Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
- Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity
- Singularity for the macroscopic production model with Chaplygin gas
- Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities
- Global dynamics of population-toxicant models with nonlocal dispersals
- α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
- High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness
- On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
- Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
- On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
- Remark on the analyticity of the fractional Fokker-Planck equation
- Continuous dependence on initial data for damped fourth-order wave equation with strain term
- Unilateral problems for quasilinear operators with fractional Riesz gradients
- Boundedness of solutions to quasilinear elliptic systems
- Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition
- Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
- Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals
- Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms
- Shape of extremal functions for weighted Sobolev-type inequalities
- One-dimensional boundary blow up problem with a nonlocal term
- Doubling measure and regularity to K-quasiminimizers of double-phase energy
- General solutions of weakly delayed discrete systems in 3D
- Global well-posedness of a nonlinear Boussinesq-fluid-structure interaction system with large initial data
- Optimal large time behavior of the 3D rate type viscoelastic fluids
- Local well-posedness for the two-component Benjamin-Ono equation
- 10.1515/anona-2025-0091
Articles in the same Issue
- Research Articles
- Incompressible limit for the compressible viscoelastic fluids in critical space
- Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
- Intervals of bifurcation points for semilinear elliptic problems
- On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
- Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
- Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
- Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
- Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
- Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
- Existence results for non-coercive problems
- Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
- Geometric characterization of generalized Hajłasz-Sobolev embedding domains
- Subharmonic solutions of first-order Hamiltonian systems
- Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
- Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
- Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
- Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
- Homoclinic solutions in periodic partial difference equations
- Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
- Properties of minimizers for L2-subcritical Kirchhoff energy functionals
- Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
- Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
- Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
- Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
- Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
- Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
- Existence of positive radial solutions of general quasilinear elliptic systems
- Low Mach number and non-resistive limit of magnetohydrodynamic equations with large temperature variations in general bounded domains
- Sharp viscous shock waves for relaxation model with degeneracy
- Bifurcation and multiplicity results for critical problems involving the p-Grushin operator
- Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
- Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
- Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity
- Singularity for the macroscopic production model with Chaplygin gas
- Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities
- Global dynamics of population-toxicant models with nonlocal dispersals
- α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
- High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness
- On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
- Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
- On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
- Remark on the analyticity of the fractional Fokker-Planck equation
- Continuous dependence on initial data for damped fourth-order wave equation with strain term
- Unilateral problems for quasilinear operators with fractional Riesz gradients
- Boundedness of solutions to quasilinear elliptic systems
- Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition
- Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
- Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals
- Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms
- Shape of extremal functions for weighted Sobolev-type inequalities
- One-dimensional boundary blow up problem with a nonlocal term
- Doubling measure and regularity to K-quasiminimizers of double-phase energy
- General solutions of weakly delayed discrete systems in 3D
- Global well-posedness of a nonlinear Boussinesq-fluid-structure interaction system with large initial data
- Optimal large time behavior of the 3D rate type viscoelastic fluids
- Local well-posedness for the two-component Benjamin-Ono equation
- 10.1515/anona-2025-0091
