Representation of solutions of systems of linear differential equations with multiple delays and linear parts given by nonpermutable matrices
In recent results on representation of solutions of systems of delayed differential equations the condition that the linear parts are given by pairwise permutable matrices was assumed. In this paper it is shown how this strong condition can be avoided, and representation of solutions of systems of d...
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Medveď, M. Pospíšil, M. 2021-02-14T08:17:58Z 2021-02-14T08:17:58Z 2016 Representation of solutions of systems of linear differential equations with multiple delays and linear parts given by nonpermutable matrices / M. Medveď, M. Pospíšil // Нелінійні коливання. — 2016. — Т. 19, № 4. — С. 521-532 — Бібліогр.: 10 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/177287 517.9 In recent results on representation of solutions of systems of delayed differential equations the condition that the linear parts are given by pairwise permutable matrices was assumed. In this paper it is shown how this strong condition can be avoided, and representation of solutions of systems of differential equations with nonconstant coefficients and variable delays is derived. The results are applied to a system with two constant delays. Also the nonexistence of blow-up solutions is proved for nonlinear systems. Останнi результати про зображення розв’язкiв систем диференцiальних рiвнянь iз запiзненням були отриманi за умови комутування матриць, що визначають лiнiйнi частини. У статтi показано як можна позбутися цiєї сильної умови та отримано зображення розв’язкiв систем диференцiальних рiвнянь з несталими коефiцiєнтами та запiзненнями, що змiнюються. Цi результати застосовано до системи зi сталими запiзненнями. Також для нелiнiйних систем доведено вiдсутнiсть вибухових розв’язкiв. en Інститут математики НАН України Нелінійні коливання Representation of solutions of systems of linear differential equations with multiple delays and linear parts given by nonpermutable matrices Зображення розв’язків систем лінійних диференціальних рівнянь з багатьма запізненнями та лінійними частинами, визначеними некомутуючими матрицями Представление решений систем линейных дифференциальных уравнений со многими запаздываниями и линейными частями, определенными некоммутирующими матрицами Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Representation of solutions of systems of linear differential equations with multiple delays and linear parts given by nonpermutable matrices |
| spellingShingle |
Representation of solutions of systems of linear differential equations with multiple delays and linear parts given by nonpermutable matrices Medveď, M. Pospíšil, M. |
| title_short |
Representation of solutions of systems of linear differential equations with multiple delays and linear parts given by nonpermutable matrices |
| title_full |
Representation of solutions of systems of linear differential equations with multiple delays and linear parts given by nonpermutable matrices |
| title_fullStr |
Representation of solutions of systems of linear differential equations with multiple delays and linear parts given by nonpermutable matrices |
| title_full_unstemmed |
Representation of solutions of systems of linear differential equations with multiple delays and linear parts given by nonpermutable matrices |
| title_sort |
representation of solutions of systems of linear differential equations with multiple delays and linear parts given by nonpermutable matrices |
| author |
Medveď, M. Pospíšil, M. |
| author_facet |
Medveď, M. Pospíšil, M. |
| publishDate |
2016 |
| language |
English |
| container_title |
Нелінійні коливання |
| publisher |
Інститут математики НАН України |
| format |
Article |
| title_alt |
Зображення розв’язків систем лінійних диференціальних рівнянь з багатьма запізненнями та лінійними частинами, визначеними некомутуючими матрицями Представление решений систем линейных дифференциальных уравнений со многими запаздываниями и линейными частями, определенными некоммутирующими матрицами |
| description |
In recent results on representation of solutions of systems of delayed differential equations the condition that the linear parts are given by pairwise permutable matrices was assumed. In this paper it is shown how this strong condition can be avoided, and representation of solutions of systems of differential equations with nonconstant coefficients and variable delays is derived. The results are applied to a system with two constant delays. Also the nonexistence of blow-up solutions is proved for nonlinear systems.
Останнi результати про зображення розв’язкiв систем диференцiальних рiвнянь iз запiзненням були отриманi за умови комутування матриць, що визначають лiнiйнi частини. У статтi показано як можна позбутися цiєї сильної умови та отримано зображення розв’язкiв систем диференцiальних рiвнянь з несталими коефiцiєнтами та запiзненнями, що змiнюються. Цi результати застосовано до системи зi сталими запiзненнями. Також для нелiнiйних систем доведено вiдсутнiсть вибухових розв’язкiв.
|
| issn |
1562-3076 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/177287 |
| citation_txt |
Representation of solutions of systems of linear differential equations with multiple delays and linear parts given by nonpermutable matrices / M. Medveď, M. Pospíšil // Нелінійні коливання. — 2016. — Т. 19, № 4. — С. 521-532 — Бібліогр.: 10 назв. — англ. |
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| first_indexed |
2025-11-25T22:45:17Z |
| last_indexed |
2025-11-25T22:45:17Z |
| _version_ |
1850570973972004864 |
| fulltext |
UDC 517.9
REPRESENTATION OF SOLUTIONS OF SYSTEMS OF LINEAR
DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS
AND LINEAR PARTS GIVEN BY NONPERMUTABLE MATRICES*
ЗОБРАЖЕННЯ РОЗВ’ЯЗКIВ СИСТЕМ ЛIНIЙНИХ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ З БАГАТЬМА ЗАПIЗНЕННЯМИ
ТА ЛIНIЙНИМИ ЧАСТИНАМИ, ВИЗНАЧЕНИМИ
НЕКОМУТУЮЧИМИ МАТРИЦЯМИ
M. Medved’
Dep. Math., Anal. and Numer. Math., Fac. Math., Phys.
and Inform., Comenius Univ. Bratislava
Mlynská dolina, 842 48 Bratislava, Slovak Republic
e-mail: Milan.Medved@fmph.uniba.sk
M. Pospı́šil
Dep. Math., Anal. and Numer. Math., Fac. Math., Phys.
and Inform., Comenius Univ. Bratislava
Mlynská dolina, 842 48 Bratislava, Slovak Republic
and Math. Inst. Slovak Acad. Sci.
Štefánikova 49, 814 73 Bratislava, Slovak Republic
e-mail: Michal.Pospisil@fmph.uniba.sk
In recent results on representation of solutions of systems of delayed differential equations the condition
that the linear parts are given by pairwise permutable matrices was assumed. In this paper it is shown how
this strong condition can be avoided, and representation of solutions of systems of differential equations
with nonconstant coefficients and variable delays is derived. The results are applied to a system with two
constant delays. Also the nonexistence of blow-up solutions is proved for nonlinear systems.
Останнi результати про зображення розв’язкiв систем диференцiальних рiвнянь iз запiзненням
були отриманi за умови комутування матриць, що визначають лiнiйнi частини. У статтi по-
казано як можна позбутися цiєї сильної умови та отримано зображення розв’язкiв систем ди-
ференцiальних рiвнянь з несталими коефiцiєнтами та запiзненнями, що змiнюються. Цi резуль-
тати застосовано до системи зi сталими запiзненнями. Також для нелiнiйних систем доведено
вiдсутнiсть вибухових розв’язкiв.
1. Introduction and preliminaries. Systems of linear differential equations with one or multi-
ple constant delays were considered in [5, 6] and solutions were represented using matrix
polynomials of a time-dependent degree. In the case of multiple delays, pairwise permutabi-
lity of matrices representing coefficients of linear terms was a necessary condition for deriving
the representation. That means that the matrices A,B1, . . . , Bn in the equation
ẋ(t) = Ax(t) +B1x(t− τ1) + . . .+Bnx(t− τn) + f(t), t ≥ 0,
∗ The first author was supported by the Slovak Research and Development Agency under the contract
No. APVV-14-0378. The second author was supported by the Grant VEGA-SAV 2/0153/16.
c© M. Medved’, M. Pospı́šil, 2016
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 4 521
522 M. MEDVED’, M. POSPÍŠIL
had to satisfyABi = BiA, BiBj = BjBi for each i, j = 1, . . . , n. For now the most general case
with nonconstant coefficients and variable delays was investigated in [9]. We recall this result.
Theorem 1.1. Let n ∈ N, Bi ∈ C([0,∞), L(RN ,RN )), gi ∈ G0 for i = 1, . . . , n and
Gs :=
{
g ∈ C([s,∞),R) | g(t) < t on [s,∞), g is increasing
}
,
f ∈ C([0,∞),RN ), γ := min{g1(0), . . . , gn(0)}, ϕ ∈ C([γ, 0],RN ). Then the solution of the
equation
ẋ(t) = B1(t)x(g1(t)) + . . .+Bn(t)x(gn(t)) + f(t), t ≥ 0, (1.1)
satisfying the initial condition
x(t) = ϕ(t), γ ≤ t ≤ 0, (1.2)
has the form
x(t) =
ϕ(t), γ ≤ t < 0,
XB1,...,Bn
g1,...,gn (t, 0)ϕ(0) +
∫ t
0
XB1,...,Bn
g1,...,gn (t, s)×
× [B1(s)ψ(g1(s)) + . . .+Bn(s)ψ(gn(s))] ds+
+
∫ t
0
XB1,...,Bn
g1,...,gn (t, s)f(s)ds, 0 ≤ t,
(1.3)
where
ψ(t) =
{
ϕ(t), t ∈ [γ, 0),
θ, t /∈ [γ, 0),
(1.4)
with the N -dimensional vector θ of zeros, and
XB1,...,Bm
g1,...,gm (t, s) :=
Θ, t < s,
Y (t, s), s ≤ t < g−1m (s),
Y (t, s) +
∫ t
g−1
m (s)
Y (t, q1)Bm(q1)Y (gm(q1), s)dq1 + . . .
. . .+
∫ t
g−km (s)
Y (t, q1)Bm(q1)×
×
∫ gm(q1)
g
−(k−1)
m (s)
Y (gm(q1), q2)Bm(q2)× . . .
. . .×
∫ gm(qk−1)
g−1
m (s)
Y (gm(qk−1), qk)×
×Bm(qk)Y (gm(qk), s)dqk . . . dq1,
g−km (s) ≤ t < g
−(k+1)
m (s), k ∈ N,
(1.5)
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 4
REPRESENTATION OF SOLUTIONS OF SYSTEMS OF LINEAR DIFFERENTIAL . . . 523
where Y (t, s) = X
B1,...,Bm−1
g1,...,gm−1 (t, s), for m = 2, . . . , n, the N ×N zero matrix Θ, and
XB
g (t, s) :=
Θ, t < s,
I, s ≤ t < g−1(s),
I +
∫ t
g−1(s)
B(q1)dq1 + . . .+
∫ t
g−k(s)
B(q1)×
×
∫ g(q1)
g−(k−1)(s)
B(q2) . . .
∫ g(qk−1)
g−1(s)
B(qk)dqk . . . dq1,
g−k(s) ≤ t < g−(k+1)(s), k ∈ N.
Note that the pairwise permutability of B1(t), . . . , Bn(t) was not needed. However, a result
from [9] on the equation involving a nondelayed term on the right-hand side was stated only
for the case of pairwise permutable constant coefficients. In the present paper we show how to
tackle this problem without permutability condition and even for nonconstant coefficients. So
this is the most general case for the delay functions from the class G0.
To illustrate our results, we consider a system of delayed differential equations with constant
coefficients and two constant delays in Section 3, and derive a formula for its solution. A known
result for permutable matrices [10] (see also [3]) is obtained as a particular case.
In the final section we apply results of Section 2 to prove a criterion on a nonexistence of
blow-up solutions for differential equations with variable coefficients and nonconstant delays.
By such a solution we understand a function x : [a, b) → RN with a ∈ R, a < b ≤ ∞ such that
limt→T− ‖x(t)‖ = ∞ for some a < T < b and a vector norm ‖ · ‖. We note that the main result
of this section (Theorem 4.1) is a generalization of a weaker result from [7].
Further applications of results of this paper can be achieved, e.g., in stability theory [6 – 9]
or Fredholm boundary-value problems [2].
2. Representation of solutions of general time dependent systems. In this section we shall
investigate systems of differential equations with multiple variable delays. For the simplicity we
start to consider the equation with constant coefficients,
ẋ(t) = Ax(t) +B1x(g1(t)) + . . .+Bnx(gn(t)) + f(t), t ≥ 0. (2.1)
Theorem 2.1. Let n ∈ N, A, Bi, i = 1, . . . , n, be N ×N matrices, gi ∈ G0 for i = 1, . . . , n,
γ := min{g1(0), . . . , gn(0)}, f ∈ C([0,∞),RN ) and ϕ ∈ C([γ, 0],RN ) be given functions. Then
the solution of the initial value problem (2.1), (1.2) has the form
x(t) =
ϕ(t), γ ≤ t < 0,
X̃(t, 0)ϕ(0) +
∫ t
0
X̃(t, s)[B1ψ(g1(s)) + . . .
. . .+Bnψ(gn(s))]ds+
∫ t
0
X̃(t, s)f(s)ds, 0 ≤ t,
for ψ given by (1.4), and
X̃(t, s) = eAtXB̃1,...,B̃n
g1,...,gn (t, s)e−As
for XB1,...,Bn
g1,...,gn (t, s) defined by (1.5), and B̃i(t) = e−AtBie
Agi(t) for i = 1, . . . , n.
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 4
524 M. MEDVED’, M. POSPÍŠIL
Proof. Let x(t) = eAty(t). Then y(t) satisfies
ẏ(t) = B̃1(t)y(g1(t)) + . . .+ B̃n(t)y(gn(t)) + f̃(t), t ≥ 0,
y(t) = ϕ̃(t), γ ≤ t ≤ 0,
(2.2)
for f̃(t) = e−Atf(t), ϕ̃(t) = e−Atϕ(t). Theorem 1.1 implies
y(t) =
ϕ̃(t), γ ≤ t < 0,
XB̃1,...,B̃n
g1,...,gn (t, 0)ϕ̃(0) +
∫ t
0
XB̃1,...,B̃n
g1,...,gn (t, s)×
×
[
B̃1(s)ψ̃(g1(s)) + . . .+ B̃n(s)ψ̃(gn(s))
]
ds+
+
∫ t
0
XB̃1,...,B̃n
g1,...,gn (t, s)f̃(s)ds, 0 ≤ t,
(2.3)
where
ψ̃(t) =
{
ϕ̃(t), t ∈ [γ, 0),
θ, t /∈ [γ, 0).
(2.4)
Note that ϕ̃(0) = ϕ(0), and ψ̃(gi(t)) = e−Agi(t)ψ(gi(t)) for i = 1, . . . , n. The statement is
obtained when one returns back to x(t).
Now, we turn to variable coefficients. So we shall consider the equation
ẋ(t) = A(t)x(t) +B1(t)x(g1(t)) + . . .+Bn(t)x(gn(t)) + f(t), t ≥ 0. (2.5)
Theorem 2.2. Let n ∈ N, A,Bi ∈ C([0,∞), L(RN ,RN )), gi ∈ G0 for i = 1, . . . , n, γ :=
:= min{g1(0), . . . , gn(0)}, f ∈ C([0,∞),RN ) and ϕ ∈ C([γ, 0],RN ) be given functions. Then
the solution of the initial value problem (2.5), (1.2) has the form
x(t) =
ϕ(t), γ ≤ t < 0,
X̃(t, 0)ϕ(0) +
∫ t
0
X̃(t, s)[B1(s)ψ(g1(s)) + . . .
. . .+Bn(s)ψ(gn(s))]ds+
∫ t
0
X̃(t, s)f(s)ds, 0 ≤ t,
(2.6)
for ψ given by (1.4), and X̃(t, s) = Φ(t)XB̃1,...,B̃n
g1,...,gn (t, s)Φ−1(s) for XB1,...,Bn
g1,...,gn (t, s) defined by (1.5),
B̃i(t) = Φ−1(t)Bi(t)Φ(gi(t)) for i = 1, . . . , n and Φ(t) is a fundamental matrix satisfying
Φ̇(t) = A(t)Φ(t), t ≥ γ,
Φ(0) = I
(2.7)
with
A(t) =
{
A(t), t ≥ 0,
A(0), t < 0,
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 4
REPRESENTATION OF SOLUTIONS OF SYSTEMS OF LINEAR DIFFERENTIAL . . . 525
where Φ̇ at 0 is considered one-sidedly.
Proof. Let x(t) = Φ(t)y(t).Then y(t) solves (2.2) with f̃(t) = Φ−1(t)f(t), ϕ̃(t) = Φ−1(t)ϕ(t)
and B̃i(t) as in the statement of the theorem. So, y(t) has the form (2.3). When one returns to
x(t) the statement immediately follows.
Remark 2.1. Note that from (1.5) it follows that XB1,...,Bn
g1,...,gn (t, s) does not depend on the
values of Bi(q) on [0, g−1i (s)) for each i = 1, . . . , n. Consequently, we can take any extension of
A(t) onto [γ,∞) instead of A(t) and the same solution x(t) given by (2.6) results.
Next, we consider a particular case of matrix functions A,B1, . . . , Bn.
Corollary 2.1. Let n ∈ N, Bi ∈ C([0,∞), L(RN ,RN )) for i = 1, . . . , n, Q ∈ C1(R, L(RN ,
RN )) be a nonsingular T -periodic matrix with Q(0) = I, R be a constant N ×N matrix, gi ∈ G0
for i = 1, . . . , n, γ := min{g1(0), . . . , gn(0)}, f ∈ C([0,∞),RN ) and ϕ ∈ C([γ, 0],RN ) be given
functions. Then the solution of the initial value problem consisting of the equation
ẋ(t) =
(
Q̇(t) +Q(t)R
)
Q−1(t)x(t) +B1(t)x(g1(t)) + . . .+Bn(t)x(gn(t)) + f(t), t ≥ 0,
and initial condition (1.2) has the form (2.6) for ψ given by (1.4), X̃(t, s) = Φ(t)XB̃1,...,B̃n
g1,...,gn (t, s)×
×Φ−1(s) for XB1,...,Bn
g1,...,gn (t, s) defined by (1.5),
B̃i(t) = Φ−1(t)Bi(t)Φ(gi(t)), i = 1, . . . , n,
and
Φ(t) =
{
Q(t)eRt, t ≥ 0,
e(Q̇(0)+R)t, t < 0,
Proof. Noting that Φ(t) is a C1 function satisfying (2.7) with
A(t) =
{ (
Q̇(t) +Q(t)R
)
Q−1(t), t ≥ 0,
Q̇(0) +R, t < 0,
the statement follows from Theorem 2.2.
3. Derivation of a formula for solutions of time independent systems. Here, we apply our
results to find a solution of a system with linear terms represented by nonpermutable constant
matrices and constant delays.
Let us consider the initial value problem consisting of the equation
ẋ(t) = B1x(t− τ1) +B2x(t− τ2) + f(t), t ≥ 0, (3.1)
and initial condition (1.2) where τ1, τ2 > 0, B1, B2 are constant N × N matrices and f ∈
∈ C([0,∞),RN ), ϕ ∈ C([γ, 0],RN ) are given functions with γ = −max{τ1, τ2}. The assumpti-
ons of Theorem 2.1 are satisfied with A = Θ and gi(t) = t− τi for i = 1, 2. Hence the theorem
gives a solution of (3.1), (1.2). In this case, X̃(t, s) = XB1,B2
g1,g2 (t, s) and the formula for the soluti-
on can be simplified. Note that XB1
g1 (·, s) is a matrix solution of
Ẋ(t) = B1X(t− τ1), t ≥ s,
X(s) = I.
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 4
526 M. MEDVED’, M. POSPÍŠIL
Thus we can use [5] to write
XB1
g1 (t, s) =
⌊
t−s
τ1
⌋∑
i=0
Bi
1
(t− iτ1 − s)i
i!
(3.2)
with the floor function b·c. Here we used the property of empty sum, i.e.,
b∑
i=a
z(i) =
∑
i∈∅
z(i) = 0
for any function z provided that a > b. We shall compute XB1,B2
g1,g2 (t, s) for s + kτ2 ≤ t <
< s+ (k + 1)τ2, k ∈ N. In the following we use the step function
σ(t) =
{
0, t < 0,
1, t ≥ 0.
Let 1 ≤ K ≤ k be an arbitrary fixed integer to extend the sums to infinity. Then
XK(t, s) :=
t∫
s+Kτ2
XB1
g1 (t, q1)B2
q1−τ2∫
s+(K−1)τ2
XB1
g1 (q1 − τ2, q2)B2 . . .
. . .
qK−1−τ2∫
s+τ2
XB1
g1 (qK−1 − τ2, qK)B2X
B1
g1 (qK − τ2, s)dqK . . . dq1 =
=
t∫
s+Kτ2
∞∑
i0=0
Bi0
1
(t− i0τ1 − q1)i0
i0!
σ(t− i0τ1 − q1)B2 . . .
. . .
qK−2−τ2∫
s+2τ2
∞∑
iK−2=0
B
iK−2
1
(qK−2 − τ2 − iK−2τ1 − qK−1)iK−2
iK−2!
×
× σ(qK−2 − τ2 − iK−2τ1 − qK−1)B2×
×
qK−1−τ2∫
s+τ2
∞∑
iK−1=0
B
iK−1
1
(qK−1 − τ2 − iK−1τ1 − qK)iK−1
iK−1!
×
× σ(qK−1 − τ2 − iK−1τ1 − qK)B2
∞∑
iK=0
BiK
1
(qK − τ2 − iKτ1 − s)iK
iK !
×
× σ(qK − τ2 − iKτ1 − s)dqK . . . dq1. (3.3)
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REPRESENTATION OF SOLUTIONS OF SYSTEMS OF LINEAR DIFFERENTIAL . . . 527
The last integral on the right-hand side is equal to
∞∑
iK−1,iK=0
B
iK−1
1 B2B
iK
1 σ(qK−1 − 2τ2 − (iK−1 + iK)τ1 − s)×
×
qK−1−τ2−iK−1τ1∫
s+τ2+iKτ1
(qK−1 − τ2 − iK−1τ1 − qK)iK−1
iK−1!
×
× (qK − τ2 − iKτ1 − s)iK
iK !
dqK =
=
∞∑
iK−1,iK=0
B
iK−1
1 B2B
iK
1
(qK−1 − 2τ2 − (iK−1 + iK)τ1 − s)iK−1+iK+1
(iK−1 + iK + 1)!
×
× σ(qK−1 − 2τ2 − (iK−1 + iK)τ1 − s) =
=
∞∑
jK−1=0
(qK−1 − 2τ2 − jK−1τ1 − s)jK−1+1
(jK−1 + 1)!
σ(qK−1 − 2τ2 − jK−1τ1 − s)×
×
jK−1∑
jK=0
B
jK−1−jK
1 B2B
jK
1 ,
where we used the substitution qK = s+ τ2 + iKτ1 + ξ(qK−1− 2τ2− (iK−1 + iK)τ1− s) leading
to a multiple of Euler beta function, and then we changed iK−1 + iK → jK−1, iK → jK .
Consequently, the last double integral on the right-hand side of (3.3) is equal to
∞∑
iK−2,jK−1=0
B
iK−2
1 B2
jK−1∑
jK=0
B
jK−1−jK
1 B2B
jK
1 σ(qK−2 − 3τ2 − (iK−2 + jK−1)τ1 − s)×
×
qK−2−τ2−iK−2τ1∫
s+2τ2+jK−1τ1
(qK−2 − τ2 − iK−2τ1 − qK−1)iK−2
iK−2!
×
× (qK−1 − 2τ2 − jK−1τ1 − s)jK−1+1
(jK−1 + 1)!
dqK−1 =
=
∞∑
iK−2,jK−1=0
B
iK−2
1 B2
jK−1∑
jK=0
B
jK−1−jK
1 B2B
jK
1 ×
× (qK−2 − 3τ2 − (iK−2 + jK−1)τ1 − s)iK−2+jK−1+2
(iK−2 + jK−1 + 2)!
×
× σ(qK−2 − 3τ2 − (iK−2 + jK−1)τ1 − s) =
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528 M. MEDVED’, M. POSPÍŠIL
=
∞∑
jK−2=0
(qK−2 − 3τ2 − jK−2τ1 − s)jK−2+2
(jK−2 + 2)!
σ(qK−2 − 3τ2 − jK−2τ1 − s)×
×
∞∑
jK−1=0
B
jK−2−jK−1
1 B2
jK−1∑
jK=0
B
jK−1−jK
1 B2B
jK
1
for jK−2 = iK−2 + jK−1. Analogously for other multiple integrals. Finally, the right-hand side
of (3.3) is equal to
∞∑
i0,j1=0
Bi0
1 B2
j1∑
j2=0
Bj1−j2
1 B2 . . .
jK−1∑
jK=0
B
jK−1−jK
1 B2B
jK
1 σ(t−Kτ2 − (i0 + j1)τ1 − s)×
×
t−i0τ1∫
s+Kτ2+j1τ1
(t− i0τ1 − q1)i0
i0!
(q1 −Kτ2 − j1τ1 − s)j1+K−1
(j1 +K − 1)!
dq1 =
=
∞∑
i0,j1=0
Bi0
1 B2
j1∑
j2=0
Bj1−j2
1 B2 . . .
jK−1∑
jK=0
B
jK−1−jK
1 B2B
jK
1 ×
× (t−Kτ2 − (i0 + j1)τ1 − s)i0+j1+K
(i0 + j1 +K)!
σ(t−Kτ2 − (i0 + j1)τ1 − s) =
=
⌊
t−Kτ2−s
τ1
⌋∑
j0=0
(t−Kτ2 − j0τ1 − s)j0+K
(j0 +K)!
×
×
j0∑
j1=0
Bj0−j1
1 B2
j1∑
j2=0
Bj1−j2
1 B2 . . .
jK−1∑
jK=0
B
jK−1−jK
1 B2B
jK
1 ,
where j0 = i0 + j1. Now changing j0 → i0, j0 − j1 → i1, . . . , jK−1 − jK → iK we get
XK(t, s) =
⌊
t−Kτ2−s
τ1
⌋∑
i0=0
(t−Kτ2 − i0τ1 − s)i0+K
(i0 +K)!
i0∑
i1=0
Bi1
1 B2
i0−i1∑
i2=0
Bi2
1 B2 . . .
. . .
i0−(i1+...+iK−1)∑
iK=0
BiK
1 B2B
i0−(i1+...+iK)
1 =
⌊
t−Kτ2−s
τ1
⌋∑
i0=0
(t−Kτ2 − i0τ1 − s)i0+K
(i0 +K)!
×
×
∑
i1,...,iK≥0
i1+...+iK≤i0
Bi1
1 B2B
i2
1 B2 . . . B
iK
1 B2B
i0−(i1+...+iK)
1
for each K ∈ N such that 1 ≤ K ≤ k and s + kτ2 ≤ t < s + (k + 1)τ2. So, we obtain the
following result.
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REPRESENTATION OF SOLUTIONS OF SYSTEMS OF LINEAR DIFFERENTIAL . . . 529
Proposition 3.1. The solution of the initial value problem (3.1), (1.2) has the form
x(t) =
ϕ(t), γ ≤ t < 0,
X(t, 0)ϕ(0) +B1
∫ τ1
0
X(t, s)ϕ(s− τ1)ds+
+B2
∫ τ2
0
X(t, s)ϕ(s− τ2)]ds+
∫ t
0
X(t, s)f(s)ds, 0 ≤ t,
(3.4)
where
X(t, s) =
⌊
t−s
τ2
⌋∑
K=0
⌊
t−Kτ2−s
τ1
⌋∑
i0=0
(t−Kτ2 − i0τ1 − s)i0+K
(i0 +K)!
×
×
∑
i1,...,iK≥0
i1+...+iK≤i0
Bi1
1 B2B
i2
1 B2 . . . B
iK
1 B2B
i0−(i1+...+iK)
1 .
Proof. From the previous arguments it follows that the solution has the form
x(t) =
ϕ(t), γ ≤ t < 0,
X(t, 0)ϕ(0) +
∫ t
0
X(t, s)[B1ψ(s− τ1)+
+B2ψ(s− τ2)]ds+
∫ t
0
X(t, s)f(s)ds, 0 ≤ t,
with
X(t, s) =
⌊
t−s
τ2
⌋∑
K=0
XK(t, s)
forXK(t, s) defined asX0(t, s) := XB1
g1 (t, s) of (3.2) and by (3.3) forK = 1, . . . ,
⌊
t− s
τ2
⌋
.Note
that X(t, s) = Θ whenever t < s. That gives formula (3.4).
In particular, we obtain the known result (see [10]).
Corollary 3.1. If B1B2 = B2B1, then the solution of (3.1), (1.2) has the form (3.4) where
X(t, s) =
∑
i,j≥0
iτ1+jτ2≤t−s
Bi
1B
j
2
(t− iτ1 − jτ2 − s)i+j
i!j!
.
Proof. Applying ∑
i1,...,iK≥0
i1+...+iK≤i0
1 =
(i0 +K)!
i0!K!
the statement follows immediately.
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530 M. MEDVED’, M. POSPÍŠIL
4. Nonexistence of blow-up solutions. The results of Section 2 may be applied, e.g., in stabi-
lity or controllability theory. In this section we illustrate the results by proving a nonexistence
of blow-up solutions.
First we recall the following estimation from [9].
Lemma 1. Let s ∈ R, n ∈ N, Bi ∈ C([s,∞), L(RN ,RN )) and gi ∈ Gs for i = 1, . . . , n.
Then
‖XB1,...,Bn
g1,...,gn (t, s)‖ ≤ exp
t∫
s
n∑
i=1
‖Bi(q)‖dq
for any t ≥ s.
Theorem 4.1. Let n ∈ N, A,Bi ∈ C([0,∞), L(RN ,RN )), gi ∈ G0 for i = 1, . . . , n, γ :=
:= min{g1(0), . . . , gn(0)}, f ∈ C([0,∞)× R(n+1)N ,RN ) and ϕ ∈ C([γ, 0],RN ) be given functi-
ons. Let x : [γ, b) → RN with 0 < b ≤ ∞ be a continuous solution of the equation
ẋ(t) = A(t)x(t) +
n∑
i=1
Bi(t)x(gi(t)) + f(t, x(t), x(g1(t)), . . . , x(gn(t))), t ≥ 0, (4.1)
satisfying the initial condition (1.2). If
‖f(t, u0, . . . , un)‖ ≤
n∑
i=0
Ri(t)ωi(‖ui‖), (t, u0, . . . , un) ∈ R× R(n+1)N
where Ri, ωi, i = 0, . . . , n, are continuous nonnegative functions defined on [0,∞), and ωi,
i = 0, . . . , n, are nondecreasing such that ω0(0) + . . .+ ωn(0) > 0 and
∞∫
0
du
ω0(u) +
∑n
i=1 ωi(2u)
= ∞,
then limt→T− ‖x(t)‖ < ∞ for all T ∈ (0, b).
Proof. Let us suppose in contrary that there exists a smallest T ∈ (0, b) such that
lim
t→T−
‖x(t)‖ = ∞.
By Theorem 2.2 and in its notation, x(t) satisfies
x(t) = X̃(t, 0)ϕ(0) +
n∑
i=1
t∫
0
X̃(t, s)Bi(s)ψ(gi(s))ds+
t∫
0
X̃(t, s)F (s)ds (4.2)
for t ≥ 0, where F (t) = f(t, x(t), x(g1(t)), . . . , x(gn(t))).
Now, since
Φ(t) = I +
t∫
0
A(s)Φ(s)ds, t ≥ 0,
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REPRESENTATION OF SOLUTIONS OF SYSTEMS OF LINEAR DIFFERENTIAL . . . 531
Gronwall lemma [4] yields
‖Φ(t)‖ ≤ exp
t∫
0
‖A(s)‖ds
, t ≥ 0,
for an induced matrix norm ‖ · ‖. Similarly,
Φ−1(t) = I−
t∫
0
Φ−1(s)A(s)ds, t ≥ 0,
i.e.,
‖Φ−1(t)‖ ≤ exp
t∫
0
‖A(s)‖ds
, t ≥ 0.
Therefore, along with Lemma 4.1,
‖X̃(t, s)‖ ≤ exp
t∫
0
‖A(q)‖dq +
t∫
s
n∑
i=1
‖B̃i(q)‖dq +
s∫
0
‖A(q)‖dq
=: M(t, s)
for any 0 ≤ s ≤ t.
Hence, denoting ‖ϕ‖ := maxγ≤t≤0 ‖ϕ(t)‖, from (4.2) one obtains
‖x(t)‖ ≤ M(t, 0)‖ϕ‖+
n∑
i=1
min{t,g−1
i (0)}∫
0
M(t, s)‖Bi(s)‖‖ϕ‖ds+
+
t∫
0
M(t, s)
(
R0(s)ω0(‖x(s)‖) +
n∑
i=1
Ri(s)ωi(‖x(gi(s))‖)
)
ds
for any t ≥ 0. Let us denote
m1 := ‖ϕ‖ max
0≤t≤T
M(t, 0) +
n∑
i=1
min{t,g−1
i (0)}∫
0
M(t, s)‖Bi(s)‖ds
,
m2 :=
max
0≤s≤t
0≤t≤T
M(t, s)
max
0≤t≤T
i=0,...,n
Ri(s), η(s) := ω0(s) +
n∑
i=1
ω(2s).
Then the last inequality implies
‖x(t)‖ ≤ m1 +m2
t∫
0
ω0(‖x(s)‖) +
n∑
i=1
ωi(‖x(gi(s))‖)ds =: z(t)
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532 M. MEDVED’, M. POSPÍŠIL
for any 0 ≤ t ≤ T. Note that the right-hand side z(t) is nondecreasing, m1 ≥ ‖ϕ‖ and
‖x(gi(t))‖ ≤ sup
0≤s≤t
‖x(gi(s))‖ ≤ sup
0≤s≤g−1
i (0)
‖x(gi(s))‖+ sup
g−1
i (0)≤s≤t
‖x(gi(s))‖ ≤ 2z(t).
Consequently,
‖x(t)‖ ≤ z(t) ≤ m1 +m2
t∫
0
ω0(z(s)) +
n∑
i=1
ωi(2z(s))ds = m1 +m2
t∫
0
η(z(s))ds
for any 0 ≤ t ≤ T. The well-known Bihari inequality [1] implies
Ω(‖x(t)‖) ≤ Ω(z(t)) ≤ Ω(m1) +m2t ≤ Ω(m1) +m2T, 0 ≤ t ≤ T,
where Ω(z) =
∫ z
0
ds
η(s)
. A contradiction follows from the limit
Ω(‖x(t)‖) →
∞∫
0
du
η(u)
= ∞, t → T−.
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Received 16.05.16
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