Construction of the General Solution of Planar Linear Discrete Systems with Constant Coefficients and Weak Delay
© J. Diblík et al. 2009
Received: 19 January 2009
Accepted: 30 March 2009
Published: 16 April 2009
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.
1.1. Preliminary Notions and Properties
where with . We will investigate only the case since the solution of (1.1) for is given by the known formula for .
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
with , . We show that the property of a system to be the system with weak delay is preserved by every nonsingular linear transformation.
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).
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.
conditions (1.13) are both necessary and sufficient.
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
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.
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)
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.
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
Since , (2.10), (2.12) yield . Then, from (2.11), we get , so either or .
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
2.1.3. The Case (2.5) of Two-Fold Real Root
From (2.10), (2.11), and (2.30), we get . Now we will analyse the two possible cases: and .
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).
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
Formulas (2.47), (2.49) can be used in the case as well. In this way, the ensuing result is proved.
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.
- (1)-dimensional if (2.2) has
two real distinct roots and
a two-fold real root, and
a two-fold real root and
- (2)dimensional if (2.2) has
two real distinct roots and
a pair of complex conjugate roots;
a two-fold real root and .
- (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)
- (b)In the case (2.4) of two complex conjugate roots, we have and the formula (2.29) uses only arbitrary parameters(253)
- (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)
The parameter cannot be seen as independent since it depends on the independent parameters and .
- (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)
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).
-dimensional if .
-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.
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 .
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  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]).
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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