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Construction of the General Solution of Planar Linear Discrete Systems with Constant Coefficients and Weak Delay
Advances in Difference Equations volume 2009, Article number: 784935 (2009)
Abstract
Planar linear discrete systems with constant coefficients and weak delay are considered. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, the space of solutions with a given starting dimension is pasted after several steps into a space with dimension less than the starting one. In a sense this situation copies an analogous one known from the theory of linear differential systems with constant coefficients and weak delay when the initially infinite dimensional space of solutions on the initial interval on a reduced interval, turns (after several steps) into a finite dimensional set of solutions. For every possible case, general solutions are constructed and, finally, results on the dimensionality of the space of solutions are deduced.
1. Introduction
1.1. Preliminary Notions and Properties
We use the following notation: for integers ,
,
, we define
where
or
are admitted, too. Throughout this paper, using notation
, we always assume
. In this paper we deal with the discrete planar systems

where is a fixed integer,
,
and
are constant
matrices, and
. Following the terminology (used, e.g., in [1, 2]), (1.1) is referred to as a nondelayed discrete system if
and as a delayed discrete system if
. Together with (1.1), we consider an initial (Cauchy) problem

where with
. We will investigate only the case
since the solution of (1.1) for
is given by the known formula
for
.
The existence and uniqueness of the solution of the initial problems (1.1) and (1.2) on are obvious. We recall that the solution
of (1.1) and (1.2) is defined as an infinite sequence

such that, for any , equality (1.1) holds.
The space of all initial data (1.2) with is obviously
-dimensional. Below we describe the fact that, among the systems (1.1), there are such systems that their space of solutions, being initially
-dimensional, on a reduced interval turns into a space having dimension less than
.
1.2. Systems with Weak Delay
We consider the system (1.1) and we look for a solution having the form ,
,
with a
. The usual procedure leads to a characteristic equation

where is the unit
matrix. Together with (1.1), we consider a system with the terms containing delays omitted

and the characteristic equation

Definition 1.1.
The system (1.1) is called a system with weak delay if the characteristic equations (1.4) and (1.6) corresponding to systems (1.1) and (1.5) are equal, that is, if for every

We consider a linear transformation

with a nonsingular matrix
. Then the discrete system for
is

with ,
. We show that the property of a system to be the system with weak delay is preserved by every nonsingular linear transformation.
Lemma 1.2.
If the system (1.1) is a system with weak delay, then its arbitrary linear nonsingular transformation (1.8) again leads to a system with the weak delay (1.9).
Proof.
It is easy to show that

holds since

and the equality

is assumed.
1.3. Necessary and Sufficient Conditions Determining the Weak Delay
In the forthcoming theorem, we give conditions, in terms of determinants, indicating whether a system is a system with weak delay or not.
Theorem 1.3.
System (1.1) is a system with weak delay if and only if the following three conditions hold simultaneously:

Proof.
We start with computing the determinant (1.4). We get

Now we see that, for (1.7) to hold, that is,

conditions (1.13) are both necessary and sufficient.
Remark 1.4.
It is easy to see that conditions (1.13) are equivalent to

1.4. Problem under Consideration
The aim of this paper is to show that the dimension of the space of all solutions, being initially equal to the dimension of the space of initial data (1.2) generated by discrete functions
, is, after several steps, reduced (on an interval of the form
with an
) to a dimension less than the initial one. In other words, we will show that the
-dimensional space of all solutions of (1.1) is reduced to a less-dimensional space of solutions on
. This problem is solved directly by explicitly computing the corresponding solutions of the Cauchy problems with each of the cases arising being considered. The underlying idea for such investigation is simple. If (1.1) is a system with weak delay, then the corresponding characteristic equation has only two eigenvalues instead of
eigenvalues in the case of systems with nonweak delay. This explains why the dimension of the space of solutions becomes less than the initial one. The final results (Theorems 2.5–2.8) provide the dimension of the space of solutions.
1.5. Auxiliary Formula
For the reader's convenience we recall one explicit formula (see, e.g., [3]) for the solutions of linear scalar discrete nondelayed equations used in this paper. We consider the first-order linear discrete nonhomogeneous equation

with and
. Then it is easy to verify that

Throughout the paper, we adopt the customary notation for the sum: where
is an integer,
is a positive integer and, "
" denotes the function considered independently of whether it is defined for indicated arguments or not.
2. Results
If (1.7) holds, then (1.4) and (1.6) have only two (and the same) roots simultaneously. In order to prove the properties of the family of solutions of (1.1) formulated in Section 1.4, we will separately discuss all the possible combinations of roots, that is, the cases of two real and distinct roots, a couple of complex conjugate roots, and, finally, a two-fold real root.
2.1. Jordan Forms of Matrix
and Corresponding Solutions of The Problem (1.1), (1.2)
It is known that, for every matrix , there exists a nonsingular matrix
transforming it to the corresponding Jordan matrix form
. This means that

where has the following possible forms, depending on the roots of the characteristic equation (1.6), that is, on the roots of

If (2.2) has two real distinct roots ,
, then

if the roots are complex conjugate, that is, with
, then

and, finally, in the case of one two-fold real root , we have either

or

The transformation transforms (1.1) into a system

with ,
,
. Together with (2.7), we consider an initial problem

with
where
is the initial function corresponding to the initial function
in (1.2).
Below we consider all four possible cases (2.3)–(2.6) separately.
We define

Assuming that the system (1.1) is a system with weak delay, the system (2.7), due to Lemma 1.2, is a system with weak delay again.
2.1.1. The Case (2.3) of Two Real Distinct Roots
In this case, we have . The necessary and sufficient conditions (1.13) for (2.7) turn into



Since , (2.10), (2.12) yield
. Then, from (2.11), we get
, so either
or
.
Theorem 2.1.
Let (1.1) be a system with weak delay and (2.2) admit two real distinct roots ,
. Then
. The solution of the initial problems (1.1) and (1.2) is
,
where
has, in the case
, the form

and, in the case , the form

Proof.
In the case considered we have and the transformed system (2.7) takes either the form


if or the form


if . We investigate only the initial problem (2.15), (2.16), (2.8) since the initial problem (2.17), (2.18), (2.8) can be examined in a similar way. From (2.16) and (2.8), we get

Then (2.15) becomes

First we solve this equation for . This means that we consider the problem

With the aid of formula (1.18), we get

Now we solve (2.20) for , that is, we consider the problem (with initial data deduced from (2.22)

Applying formula (1.18) yields (for )

Picking up all particular cases (2.8), (2.22), and (2.24), we have

Now, taking into account (2.9), the formula (2.13) is a consequence of (2.19) and (2.25). The formula (2.14) can be proved in a similar way.
Finally, we note that both formulas (2.13) and (2.14) remain valid for as well. In this case, the transformed system (2.7) reduces to a system without delay.
2.1.2. The Case (2.4) of Two Complex Conjugate Roots
The necessary and sufficient conditions (1.13) for (2.7) take the forms (2.10) and (2.11) and

The system of conditions (2.10), (2.11), and (2.26) gives ,
and admits only one possibility, namely,

Consequently, and
as well. The initial problems (1.1) and (1.2) reduces to a problem without delay

and, obviously,

2.1.3. The Case (2.5) of Two-Fold Real Root
We have . The necessary and sufficient conditions (1.13) are, for (2.7), reduced to (2.10), (2.11), and

From (2.10), (2.11), and (2.30), we get . Now we will analyse the two possible cases:
and
.
The Case

Theorem 2.2.
Let (1.1) be a system with weak delay, (2.2) admit a two-fold root ,
and the matrix
has the form (2.5). Then the solution of the initial problems (1.1) and (1.2) is
,
where
has, in the case
, the form

and, in the case , the form

Proof.
The assumption or
leads to
. Then the following cases arise. Either
,
or
,
or
. The latter case is covered by the above formulas (2.31) and (2.32) since it can be treated as system (2.28) considered previously (with
) when
, and the corresponding solution is described by the formula (2.29). If
, then (2.7) turns into the system

and, if , then (2.7) turns into the system

System (2.33) can be solved in much the same way as the systems (2.15) and (2.16) if we put , and the discussion of the system (2.34) copies the discussion of the systems (2.17) and (2.18) with
. Formulas (2.31) and (2.32) are consequences of (2.13) and (2.14).
The Case
For we define

Theorem 2.3.
Let the system (1.1) be a system with weak delay, (2.2) admit two repeated roots ,
, and the matrix
has the form (2.5). Then the solution of the initial problems (1.1) and (1.2) is given by
,
where
has the form

Proof.
In this case, all the entries of are nonzero and, from (2.10), (2.11), and (2.30), we get

Then the system (2.7) reduces to


where . It is easy to see (multiplying (2.39) by
and summing both equations) that

We can see (2.40) as a homogeneous equation with respect to the unknown expression . Then, using (1.18), we obtain

With the aid of (2.41), we rewrite the systems (2.38) and (2.39) as follows:

It is easy to see that the system (2.42) is decomposed into two separate equations. Solving each of them in a similar way as in the proof of Theorem 2.1 using (1.18) (details are omitted), we conclude

Formula (2.36) is now a direct consequence of (2.43) and (2.35).
2.1.4. The Case (2.6) of Two-Fold Real Root
If the matrix has the form (2.6), the necessary and sufficient conditions (1.13), for (2.7), are reduced to (2.10), (2.11), and

Then (2.10), (2.11), and (2.44) give , and the system (2.7) can be written as


Solving (2.46), we get

Then (2.45) turns into

Equation (2.48) can be solved in a similar way as in the proof of Theorem 2.1 using (1.18) (we omit details). We get

Formulas (2.47), (2.49) can be used in the case as well. In this way, the ensuing result is proved.
Theorem 2.4.
Let (1.1) be a system with weak delay, (2.2) admit two repeated roots , and the matrix
has the form (2.6). Then
and the solution of the initial problems (1.1) and (1.2) is
,
where
,
is defined by (2.49) and
by (2.47).
2.2. Dimension of the Set of Solutions
Since all the possible cases of the planar system (1.1) with weak delay have been analysed, we are ready to formulate results concerning the dimension of the space of solutions of (1.1) assuming that initial conditions (1.2) are variable.
Theorem 2.5.
Let (1.1) be a system with weak delay, and (2.2) has both roots different from zero. Then the space of solutions, being initially -dimensional, becomes on
only
-
(1)
-dimensional if (2.2) has
-
(a)
two real distinct roots and
-
(b)
a two-fold real root,
and
-
(c)
a two-fold real root and
-
(a)
-
(2)
dimensional if (2.2) has
-
(a)
two real distinct roots and
-
(b)
a pair of complex conjugate roots;
-
(c)
a two-fold real root and
.
-
(a)
Proof.
We will carefully trace all theorems considered (Theorems 2.1–2.4) together with the case of a pair of complex conjugate roots uncovered by a theorem and our conclusion will hold on (some of the statements hold on a greater interval).
-
(a)
Analysing the statement of Theorem 2.1 (the case (2.3) of two real distinct roots) we obtain the following subcases.
-
(a1)
If
,
, then the dimension of the space of solutions on
equals
since the last line in (2.13) uses only
arbitrary parameters
(250) -
(a2)
If
,
, then the dimension of the space of solutions on
equals
since the last line in (2.14) uses only
arbitrary parameters
(251) -
(a3)
If
, then the dimension of the space of solutions on
equals
since the last line in (2.13) and in (2.14) uses only
arbitrary parameters
(252)
-
(a1)
This means that all the cases considered are covered by conclusions (1a) and (2a) of Theorem 2.5.
-
(b)
In the case (2.4) of two complex conjugate roots, we have
and the formula (2.29) uses only
arbitrary parameters
(253)
for every . This is covered by case
of Theorem 2.5.
-
(c)
Analysing the statement of Theorems 2.2 and 2.3 (the case (2.5) of two-fold real root), we obtain the following subcases.
-
(c1)
If
,
, then the dimension of the space of solutions on
equals
since the last line in (2.31) uses only
arbitrary parameters
(254) -
(c2)
If
,
, then the dimension of the space of solutions on
equals
since the last line in (2.32) uses only
arbitrary parameters
(255) -
(c3)
If
then the dimension of the space of solutions on
equals
since the last line in (2.31) and in (2.32) uses only
arbitrary parameters
(256) -
(c4)
If
, then the dimension of the space of solutions on
equals
since the last line in (2.36) uses only
arbitrary parameters
(257)
-
(c1)
where

The parameter cannot be seen as independent since it depends on the independent parameters
and
.
All the cases considered are covered by conclusions (1b), (1c), and (2c) of Theorem 2.5.
-
(d)
Analysing the statement of Theorem 2.4 (The case (2.6) of two-fold real root), we obtain the following subcases.
-
(d1)
If
,
, then the dimension of the space of solutions on
equals
since the last line in (2.49) uses only
arbitrary parameters
(259)and the last line in (2.47) provides no new information.
-
(d2)
If
, then the dimension of the space of solutions on
equals
since, as follows from (2.49) and (2.47), there are only
arbitrary parameters
(260)
-
(d1)
Both cases are covered by conclusions (1b) and (2c) of Theorem 2.5.
Since there are no cases other than the above cases (a)–(d), the proof is finished.
Theorem 2.5 can be formulated simply as follows.
Theorem 2.6 (Main result).
Let (1.1) be a system with weak delay and let (2.2) have both roots different from zero. Then the space of solutions, being initially -dimensional, is on
only
-
(1)
-dimensional if
.
-
(2)
-dimensional if
.
We omit the proofs of the following two theorems since again, they can be done in much the same way as Theorems 2.1–2.4.
Theorem 2.7.
Let (1.1) be a system with weak delay and let (2.2) have a simple root . Then the space of solutions, being initially
-dimensional, is either
-dimensional or
-dimensional on
.
Theorem 2.8.
Let (1.1) be a system with weak delay and let (2.2) have a two-fold root . Then the space of solutions, being initially
-dimensional, turns into
-dimensional space on
, namely, into the zero solution.
3. Concluding Remarks
To our best knowledge, weak delay was first defined in [4] for systems of linear delayed differential systems with constant coefficients. Nevertheless, separate particular examples can be found in various books concerning delayed differential equations. Let us summarize the advantage of investigating "weak" delayed systems in the plane. Such systems can be simplified and their solutions can be found in a simple explicit analytical form. In the case of ordinary differential systems with delay, to obtain the corresponding eigenvalues, it is sufficient to solve only a polynomial equation rather than a quasipolynomial one. In the case of discrete systems of two equations investigated in this paper in the "weak" case, to obtain the corresponding eigenvalues, it is sufficient to solve only polynomial equation of the second order rather than a polynomial equation of th order. Note that results obtained can be directly used to investigate such asymptotic problems as boundedness or convergence of solutions (using different methods, such problems have recently been investigated, e.g., in [5–11]).
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Acknowledgments
The first author was supported by the Grant 201/07/0145 of Czech Grant Agency (Prague), by the Council of Czech Government MSM 00216 30503 and MSM 00216 30519. The third author was supported by the Council of Czech Government MSM 00216 30503 and MSM 00216 30529.
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Diblík, J., Khusainov, D.Y. & Šmarda, Z. Construction of the General Solution of Planar Linear Discrete Systems with Constant Coefficients and Weak Delay. Adv Differ Equ 2009, 784935 (2009). https://doi.org/10.1155/2009/784935
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DOI: https://doi.org/10.1155/2009/784935
Keywords
- Characteristic Equation
- Differential System
- Constant Coefficient
- Discrete System
- Initial Problem