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 twofold 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 twofold 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 TwoFold 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 twofold 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 TwoFold 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 twofold real root, and

(c)
a twofold real root and

(2)
dimensional if (2.2) has

(a)
two real distinct roots and

(b)
a pair of complex conjugate roots;

(c)
a twofold real root and .
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

(a2)
If , , then the dimension of the space of solutions on equals since the last line in (2.14) uses only arbitrary parameters

(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
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
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 twofold 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

(c2)
If , , then the dimension of the space of solutions on equals since the last line in (2.32) uses only arbitrary parameters

(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

(c4)
If , then the dimension of the space of solutions on equals since the last line in (2.36) uses only arbitrary parameters
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 twofold 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
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
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 twofold root . Then the space of solutions, being initially dimensional, turns into dimensional space on , namely, into the zero solution.