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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 627295, 37 pages
General Explicit Solution of Planar Weakly Delayed Linear Discrete Systems and Pasting Its Solutions
1Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic
2Department of Mathematics, Faculty of Electrical Engineering, Brno University of Technology, 616 00 Brno, Czech Republic
Received 3 September 2013; Accepted 21 October 2013; Published 29 April 2014
Academic Editor: Miroslava Růžičková
Copyright © 2014 Josef Diblík and Hana Halfarová. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Planar linear discrete systems with constant coefficients and delays are considered where , are constant integer delays, , are constant matrices, and . It is assumed that the considered system is weakly delayed. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and special delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained.
1.1. Preliminary Notions and Properties
We use the following notation: for integers , , , we define , where or is admitted, too. Throughout this paper, using notation , we always assume . In the paper, we deal with the discrete planar system where are constant integer delays, , , are constant 2 × 2 matrices, , , , , and . Throughout the paper, we assume that where and is 2 × 2 zero matrix. Together with (1), we consider an initial (Cauchy) problem where with . The existence and uniqueness of the solution of the initial problem (1), (3) on are obvious. We recall that the solution of (1), (3) is defined as an infinite sequence such that, for any , equality (1) holds.
The space of all initial data (3) with is obviously -dimensional. Below, we describe the fact that, among system (1), there are such systems that their space of solutions, being initially -dimensional, on a reduced interval turns into a space having a dimension less than . The problem under consideration (pasting property of solutions) is exactly formulated in Section 1.4.
1.2. Weakly Delayed Systems
We consider system (1) and look for a solution having the form , where , with , and is a nonzero constant vector. The usual procedure leads to a characteristic equation where is the unit 2 × 2 matrix. Together with (1), we consider a system with the terms containing delays omitted: and its characteristic equation
Definition 1. System (1) is called a weakly delayed system if characteristic equations (5), (7) corresponding to systems (1) and (6) are equal, that is, if, for every , We consider a linear transformation with a nonsingular 2 × 2 matrix , then the discrete system for is with , , where . We show that a system’s property of being one weakly delayed is preserved by every nonsingular linear transformation.
Proof. It is easy to show that holds since that is, equality (8) is assumed.
1.3. Necessary and Sufficient Conditions Determining Weakly Delayed Systems
In the next theorem, we give conditions, in terms of determinants, indicating whether a system is weakly delayed.
Theorem 3. System (1) is a weakly delayed system if and only if the following conditions hold simultaneously: where and .
Proof. We start with computing determinant defined by (5). We get
Expanding the determinant on the right-hand side along summands of the first column, we get
Expanding each of the above determinants along summands of the second column, we have
After simplification, we get Now we see that for (8) to hold; that is, conditions (13)–(16) are both necessary and sufficient.
Proof. (I) We show that assumptions (13)–(16) imply (23)–(25). It is obvious that condition (23) is equivalent to (13), (14). Now we consider
Expanding the determinant on the right-hand side along summands of the first column and then expanding each of the determinants along summands of the second column, we have
Now we consider Expanding the determinant on the right-hand side along summands of the first column and then expanding each of the determinants along summands of the second column, we have
(II) Now we prove that assumptions (23)–(25) imply (13) and (16). Due to equivalence of (13) and (14) with (23), it remains to be shown that (23)–(25) imply (15) and (16).
If (24) holds, then, from computations in (27), we see that and because of (23) we get (15).
Finally, we show that (23) and (25) imply (16). From (29) (using (23)) we get that is, (16) holds.
1.4. Problem under Consideration
The aim of this paper is to give explicit formulas for solutions of weakly delayed systems and to show that, after several steps, the dimension of the space of all solutions, being initially equal to the dimension of the space of initial data (3) generated by discrete functions , is reduced to a dimension less than the initial one on an interval of the form with an . In other words, we will show that the -dimensional space of all solutions of (1) is pasted 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) is a weakly delayed system, then the corresponding characteristic equation has only two eigenvalues instead of eigenvalues in the case of systems with nonweak delays. This explains why the dimension of the space of solutions becomes less than the initial one. The final results (Theorems 10–13) provide the dimension of the space of solutions. Our results generalize the results in [1, 2], where system (1) with and was analyzed.
1.5. Auxiliary Formula
For the reader’s convenience, we recall one explicit formula (see, e.g., ) for the solutions of linear scalar discrete nondelayed equations used in this paper. We consider initial-value problem for the first order linear discrete nonhomogeneous equation with and . Then, it is easy to verify that unique solution of this problem is 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.
Note that the formula (33) is used many times in recent literature to analyze asymptotic properties of solutions of various classes of difference equations, including nonlinear equations. We refer, for example, to [4–8] and to relevant references therein.
2. General Solution of Weakly Delayed System
If (8) holds, then (5) and (7) have only two (and the same) roots simultaneously. In order to prove the properties of the family of solutions of (1) formulated in the introduction, we will discuss each combination of roots, that is, the cases of two real and distinct roots, a pair of complex conjugate roots, and, finally, a double real root.
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 four possible forms (denoted below as ), depending on the roots of the characteristic equation (7), that is, on the roots of If (35) has two real distinct roots , , then if the roots are complex conjugate, that is, with , then and, finally, in the case of one double real root , we have either or The transformation transforms (1) into a system with , , , and . Together with (40), we consider an initial problem with where is the initial function corresponding to the initial function in (3).
2.1.1. Case (36) of Two Real Distinct Roots
In this case, we have and . The necessary and sufficient conditions (13)–(16) for (40) turn into Since , (43) and (45) yield , then, from (44), we get , so that either or . In view of assumptions , , we conclude that only the following cases I, II are possible:(I), , ,(II), , .
In Theorem 5 both cases I, II are analyzed.
Theorem 5. Let (1) be a weakly delayed system and (35) has two real distinct roots , . If case (I) holds, then the solution of the initial problem (1), (3) is , , where has the form If case (II) is true, then the solution of initial problem (1), (3) is , , where has the form
Proof. If case (I) is true, then the transformed system (40) takes the form
and if case (II) holds, then (40) takes the form
We investigate only the initial problem (49), (50), (41) since the initial problem (51), (52), (41) can be examined in a similar way.
From (50), (41), we get then (49) becomes First, we solve this equation for . This means that we consider the problem With the aid of formula (33), we get Now we solve (54) for with initial data deduced from (56); that is, we consider the problem Applying formula (33) we get (for ) Now we solve (54) for with initial data deduced from (58); that is, we consider the problem Applying formula (33) yields (for )
From (56), (58), and (60) we deduce that expected form of the solution of the initial problem for with initial data derived from the solution of previous equation for is
We solve (54) for with initial data deduced from (61); that is, we consider the problem
Applying formula (33) yields (for )
In the end we solve (54) for with initial data deduced from (63); that is, we consider the problem
Applying formula (33) yields (for )
Summing up all particular cases (56)–(65) we have