- Open Access
Explicit general solution of planar linear discrete systems with constant coefficients and weak delays
© Diblík and Halfarová; licensee Springer 2013
- Received: 7 December 2012
- Accepted: 14 February 2013
- Published: 6 March 2013
In this paper, planar linear discrete systems with constant coefficients and two delays
are considered where , , are fixed integers and , and are constant matrices. It is assumed that the considered system is one with weak delays. 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 weak 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.
AMS Subject Classification:39A06, 39A12.
- discrete equation
- weak delays
- explicit solution
- dimension of the solutions space
Preliminary notions and properties
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 systems (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 .
Systems with weak delays
with , , . We show that a system’s property of being one with weak delays is preserved by every nonsingular linear transformation.
Lemma 1 If system (1) is a system with weak delays, then its arbitrary linear nonsingular transformation (8) again leads to a system with weak delays (9).
is assumed. □
Necessary and sufficient conditions determining the weak delays
In the below theorem, we give conditions, in terms of determinants, indicating whether a system is one with weak delays.
conditions (10)-(16) are both necessary and sufficient. □
Now we prove that assumptions (17) and (21) imply (10) and (16). Due to equivalence (10)-(13) with (17) and (19), it remains to be shown that (18), (20) and (21) imply (14)-(16).
i.e., because of (17), we get (14). Similarly, formula (15) can be proved with the aid of (20) and by (19).
i.e., (16) holds. □
Problem under consideration
The aim of this paper is 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 φ, 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 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 arising cases being considered. The underlying idea for such investigation is simple. If (1) is a system with weak delays, then the corresponding characteristic equation has only two eigenvalues instead of eigenvalues in the case of systems with non-weak delays. This explains why the dimension of the space of solutions becomes less than the initial one. The final results (Theorems 7-10) provide the dimension of the space of solutions. Our results generalize the results in  where system (1) with is analyzed.
Throughout the paper, we adopt the customary notation for the sum: , where ℓ is an integer, s is a positive integer and ‘ℱ’ denotes the function considered independently of whether it is defined for indicated arguments or not.
Note that formula (24) is many times used in recent literature to analyze asymptotic properties of solutions of various classes of difference equations, including nonlinear equations. We refer, e.g., to [3–9] and to relevant references therein.
If (7) holds, then equations (4) and (6) 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, i.e., the cases of two real and distinct roots, a pair of complex conjugate roots and, finally, a double real root.
Jordan forms of matrix A and corresponding solutions of problem (1), (3)
with , where is the initial function corresponding to the initial function φ in (3).
Below, we consider all four possible cases (26)-(29) separately.
Assuming that (1) is a system with weak delays, by Lemma 1, system (30) is one with weak delays again.
The case (26) of two real distinct roots
Now, taking into account (32), formula (40) is a consequence of (46) and (51). Formula (41) can be proved in a similar way.
Finally, we note that both formulas (40), (41) remain valid for , as well. In this case, the transformed system (1) reduces to a system without delays. This possibility is excluded by conditions (2). □
The case (27) of two complex conjugate roots
From this discussion, the next theorem follows.
Theorem 3 There exists no system (1) with weak delays if Λ has the form (27).
Finally, we note that assumptions (2) alone exclude this case.
The case (28) of double real root
Now we will analyze the two possible cases: and .
For the case , we have from (33), (35) that and or . For and , condition (37) gives . Then, from (34), (36), we get and .
Now we discuss the case . From conditions (33), (35), we have and . This yields , , and from (60), we have , . By conditions (34), (36), we get , .
The case ,
System (63), (64) can be solved in much the same way as system (42), (43) if we put , and the discussion of system (65), (66) goes along the same lines as that of system (44), (45) with . Formulas (61) and (62) are consequences of (40), (41). □
The case ,
Formula (69) is now a direct consequence of (82), (83) and (67), (68). □
The case (29) of a double real root
Then (33), (35) and (84) give , and from (34), (36) and (85), we have .