Positive solutions of linear impulsive differential equations
The paper deals with existence of positive (nonnegative) solutions of linear homogeneous impulsive differential equations. The main result is also applied to investigate the similar problem for higher order linear homogeneous impulsive differential equations. All results are formulated in terms of...
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Інститут математики НАН України
2005
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irk-123456789-1780102021-02-19T01:26:54Z Positive solutions of linear impulsive differential equations Akhmet, M.U. Yilmaz, O. The paper deals with existence of positive (nonnegative) solutions of linear homogeneous impulsive differential equations. The main result is also applied to investigate the similar problem for higher order linear homogeneous impulsive differential equations. All results are formulated in terms of coefficients of the equations. Розглядається iснування додатних (невiд’ємних) розв’язкiв лiнiйних однорiдних диференцiальних рiвнянь з iмпульсною дiєю. Основний результат також використовується для вивчення подiбної проблеми для лiнiйних однорiдних рiвнянь вищого порядку з iмпульсною дiєю. Всi результати сформульовано в термiнах коефiцiєнтiв рiвнянь. 2005 Article Positive solutions of linear impulsive differential equations / M.U. Akhmet, O. Yilmaz // Нелінійні коливання. — 2005. — Т. 8, № 3. — С. 291-297. — Бібліогр.: 13 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/178010 517.9 en Нелінійні коливання Інститут математики НАН України |
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The paper deals with existence of positive (nonnegative) solutions of linear homogeneous impulsive differential equations. The main result is also applied to investigate the similar problem for higher order
linear homogeneous impulsive differential equations. All results are formulated in terms of coefficients of
the equations. |
| format |
Article |
| author |
Akhmet, M.U. Yilmaz, O. |
| spellingShingle |
Akhmet, M.U. Yilmaz, O. Positive solutions of linear impulsive differential equations Нелінійні коливання |
| author_facet |
Akhmet, M.U. Yilmaz, O. |
| author_sort |
Akhmet, M.U. |
| title |
Positive solutions of linear impulsive differential equations |
| title_short |
Positive solutions of linear impulsive differential equations |
| title_full |
Positive solutions of linear impulsive differential equations |
| title_fullStr |
Positive solutions of linear impulsive differential equations |
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Positive solutions of linear impulsive differential equations |
| title_sort |
positive solutions of linear impulsive differential equations |
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Інститут математики НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/178010 |
| citation_txt |
Positive solutions of linear impulsive differential equations / M.U. Akhmet, O. Yilmaz // Нелінійні коливання. — 2005. — Т. 8, № 3. — С. 291-297. — Бібліогр.: 13 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT akhmetmu positivesolutionsoflinearimpulsivedifferentialequations AT yilmazo positivesolutionsoflinearimpulsivedifferentialequations |
| first_indexed |
2025-07-15T16:18:59Z |
| last_indexed |
2025-07-15T16:18:59Z |
| _version_ |
1837730459886288896 |
| fulltext |
UDC 517 . 9
POSITIVE SOLUTIONS OF LINEAR IMPULSIVE
DIFFERENTIAL EQUATIONS
ДОДАТНI РОЗВ’ЯЗКИ ЛIНIЙНИХ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ З IМПУЛЬСНОЮ ДIЄЮ
M. U. Akhmet
Middle East Techn. Univ.
06531 Ankara, Turkey
O. Yilmaz
Abant Izzet Baysal Univ.
Gölköy Kampüsü, 14280, Bolu, Turkey
The paper deals with existence of positive (nonnegative) solutions of linear homogeneous impulsive dif-
ferential equations. The main result is also applied to investigate the similar problem for higher order
linear homogeneous impulsive differential equations. All results are formulated in terms of coefficients of
the equations.
Розглядається iснування додатних (невiд’ємних) розв’язкiв лiнiйних однорiдних диференцiаль-
них рiвнянь з iмпульсною дiєю. Основний результат також використовується для вивчення
подiбної проблеми для лiнiйних однорiдних рiвнянь вищого порядку з iмпульсною дiєю. Всi ре-
зультати сформульовано в термiнах коефiцiєнтiв рiвнянь.
1. Introduction and preliminaries. The problem of positive solutions for different type of di-
fferential equations has attracted the attention of many researchers [1 – 4]. They considered
positiveness of solutions defined on the positive half line, whole line, and solutions of boundary-
value problems.
In the last several decades, theory of impulsive differential equations has been developed
very intensively to keep up with demands of disciplines such as biology, mechanics, medicine,
etc. Many results of the theory can be found in profoundly written books [5] and [6] and in the
references cited in these books. Naturally, the problem of positive solutions for the impulsive
differential equations has become important. One can mention the paper [7] in this subject.
Our article concerns with existence of positive solutions of linear impulsive homogeneous
systems of the first order and of higher order linear equations. Apparently, this work is one of
the first in the subject.
To obtain the result we shall develop, for impulsive differential equations, a method which
was first proposed in [8]. We intent to find conditions on the impulsive part of the systems, which
provide, together with conditions on differential equations, existence of nonnegative solutions
on positive half line.
Let us denote by R, N, the sets of all real numbers and positive integers respectively. Throug-
hout the paper, some abbreviations are used to simplify the notation: If x = (x1, . . . , xn) is a
vector, then x ≥ 0 means that xk ≥ 0 for k = 1, . . . , n. Similarly, if A = (aik) is an n×n matrix,
then A ≥ 0 means that aik ≥ 0 for i, k = 1, . . . , n.
c© M. U. Akhmet, O. Yilmaz, 2005
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3 291
292 M. U. AKHMET, O. YILMAZ
Consider the system of impulsive linear differential equations,
x′(t) = −A(t)x, t 6= θi, (1)
∆x |t=θk
= −Bkx, (2)
where x ∈ Rn, t ∈ R+ = [0,∞), A(t) is an n × n matrix of continuous functions and Bk are
constant n× n matrices.
We assume that throughout the paper, the following conditions on the system (1), (2) are
fulfilled:
C1) A(t) ∈ C(R+), A(t) ≥ 0 for t ∈ R+;
C2) {θk} ⊂ R+\{0}, k ∈ N, is a strictly ordered sequence such that θk → ∞ as k → ∞;
C3) Bk ≥ 0, det(I −Bk)−1 6= 0 and (I −Bk)−1 ≥ 0 for k ∈ N.
From the theory of impulsive differential equations [5, 6], it is known that the system (1),
(2) satisfies the conditions for existence and uniqueness of solution. Moreover, every solution
x(t) of the system can be continued to +∞.
We also consider the linear impulsive differential equation of n-th order,
a0(t)y(n) +
n∑
k=1
(−1)k+1ak(t)y(n−k) = 0, t 6= θi, (3)
∆ŷ |t=θi
= Bŷ, (4)
where ŷ(t) = [y(t), . . . , y(n−1)(t)]T , y(i) =
diy
dti
and
Bk =
bk
11 bk
12 . . . bk
1n
bk
21 bk
22 . . . bk
2n
. . . . . . . . . . . .
bk
n1 bk
n2 . . . bk
nn
.
For the n-th order system, following conditions are imposed:
D1) the coefficient functions ak(t) ∈ C(R+) for k = 1, . . . , n satisfy the following:
a0 > 0, ak ≥ 0, k = 2, . . . , n;
D2) {θk} ⊂ R+\{0}, k ∈ N, is a strictly ordered sequence such that θk → ∞ as k → ∞;
D3) (−1)i+jbij ≤ 0 for i, j = 1, . . . , n, det(I −Bk)−1 6= 0 and (I −Bk)−1 ≥ 0, k ∈ N.
One should emphasize that the idea of considering higher order impulsive differential equati-
ons with discontinuity in all derivatives is not new [9 – 13]. It is understood that the solutions
of the systems (1), (2) and (3), (4) are left continuous functions with discontinuities of the first
type at the points θi.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
POSITIVE SOLUTIONS OF LINEAR IMPULSIVE DIFFERENTIAL EQUATIONS 293
2. Main results.
Theorem 1. Assume that assertions C1) – C3) are fulfilled. Then there exists a solution x =
= x(t) of (1), (2), which is not identically zero, satisfying
x(t) ≥ 0, −x′(t) ≥ 0, t ∈ R+, (5)
−∆x(θi) ≥ 0, ∆x′(θi) ≥ 0, i ∈ N. (6)
Proof. We intend to prove existence of positive valued solution x0(t) constructively as a
limit of solutions which are positive on sections.
a) Fix an integer r > 0 and x0 > 0. We shall show that for a given initial condition (r, x0),
there exists a positive valued solution xr(t) = x(t, r, x0) of (1) and (2) on [0, r]. We assume,
without loss of generality, that θl < r ≤ θl+1 for some l ≥ 1. Let us show that on (θl, r], xr(t)
is decreasing and positive valued. Since xr(r) = x0 > 0, it follows that xr(t) > 0 for t near
r, hence
xr(t)
dt
≤ 0 for t near r. Thus xr(t) ≥ xr(r) > 0 for t less than and close to r. This
argument shows that xr(t) > 0 and
xr(t)
dt
≤ 0 for t ∈ (θl, r]. In order to define a value xr(θ−l ),
we, by using (2), obtain,
xr(θ+
l ) = (I −Bl)xr(θ−l )
and, hence,
xr(θ−l ) = (I −Bl)−1xr(θ+
l ).
Then, we could continue the solution on (θl−1, θl]. Proceeding in this way, we construct the
solution xr(θi) on interval [0, r]. In the case when 0 < r < θ1 one can proceed in the same
manner as it has been done for the interval (θl, r] above.
b) In stage a) we construct a solution xr(t) for every r ≥ 1. It is clear that zr(t) =
xr(t)
‖xr(0)‖
is a solution of (1) and (2) and ‖zr(0‖ = 1. There exists a subsequence of zr(0), r ≥ 1, which
converges to z0 with ‖z0‖ = 1 (we assume without loss of generality that the convergent
subsequence is the sequence zr(0) itself). Fix an arbitrary integer i ≥ 1. Using Theorem 5
from [5], one can show that zr(t) is convergent in sup-norm on [0, i] to the function x0(t), which
is a solution of (1) and (2) on the interval [0, i] and it is nonnegative since all solutions zr(t)
are positive on [0, i] if r ≥ i. Since i is arbitrary, x0(t) is a nonnegative solution of (1) and (2)
on R+.
c) Substituting x0(t) in (1) we obtain the second inequality in (5),
dx0(t)
dt
= −A(t)x0(t) ≤ 0, t 6= θi.
Similarly using x0(θi) ≥ 0, condition C3) and equation (2), we have the first inequality in (6) as
follows:
−∆x0(θi) = Bi x
0(θi) ≥ 0.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
294 M. U. AKHMET, O. YILMAZ
While for the second inequality in (6), we use (2) again,
∆x0′
(θi) = x0′
(θ+
i )− x0′
(θ−i ) = −A(θ+
i ) x(θ+
i ) + A(θ−i ) x(θ−i ) = −A(θi) ∆x(θi) ≥ 0.
This concludes the proof.
In the following theorem, we generalize the above result to the n-th order case.
Theorem 2. Assume that the linear impulsive differential equation of n-th order satisfies the
conditions D1) – D3). Then the system (3), (4) has a solution y = y(t) which is positive for
t ∈ R+ and, what is more,
(−1)j y(j) ≥ 0, j = 0, 1, . . . , n− 1, (7)
∆y ≤ 0,∆y′ ≥ 0, . . . , (−1)n−1∆y(n−1) ≤ 0. (8)
Proof. Let
g(t) = exp
t∫
1
a1(s)ds
> 0.
Next, we change the variables,
y = x1, y
′
= −x2, . . . , y
(n−2) = (−1)n−2xn, y(n−1) = (−1)n−1 xn
g
.
Then, the system becomes
x′1 = −x2, t 6= θi,
x′2 = −x3, t 6= θi,
. . . . . . . . . . . . . . . . . .
x′n−1 = −xn
g
, t 6= θi,
x′n = −gxn−1
a2(t)
a0(t)
− gxn−2
a3(t)
a0(t)
− . . .− gx1
an(t)
a0(t)
, t 6= θi,
∆x̂ |t=θi
= B̂ix̂,
where x̂(t) = [x1(t), . . . , xn(t)]T and
B̂i =
bi
11 −bi
12 · · · (−1)n+1bi
1n
−bi
21 bi
22 · · · (−1)n+2bi
2n
. . . . . . . . . . . .
(−1)n+1bi
n1 (−1)n+2bi
n2 . . . (−1)n+nbi
nn
.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
POSITIVE SOLUTIONS OF LINEAR IMPULSIVE DIFFERENTIAL EQUATIONS 295
If the above transformed system is identified with the one in Theorem 1, we have
x̂(t) ≥ 0, −x̂ ′(t) ≥ 0, and −∆x̂ ′(t) ≥ 0, ∆x̂ ′(t) ≥ 0.
By retaining the original variables we have
y
−y′
...
(−)n−1yn−1
≥ 0 and −
∆y
−∆y′
...
(−1)n−1∆yn−1
≥ 0.
This concludes the proof.
3. Examples.
Example 1. Consider the following coupled system:
mx′′ = 2kx + ky, t 6= θi,
2my′′ = kx + 2ky, t 6= θi,
∆x|t=θi
= −1
2
x +
1
4
x′,
(9)
∆x′|t=θi
=
1
4
x− 1
2
x′,
∆y|t=θi
= −1
2
y,
∆y′|t=θi
= −1
2
y′,
where m and k are positive real numbers and θi = 2i, i = 1, 2, . . . . Theorem 2 is not appli-
cable for this example. But by changing the variables in the above system, Theorem 1 will be
applicable,
x = z1, y = z3, x′ = −z2, y′ = −z4
so, the system becomes,
z′ = −Az, t 6= θi,
∆z|t=θi
= Bz,
where z = [z1, z2, z3, z4]T . If the above system is identified with (1), (2), the matrices A and B
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
296 M. U. AKHMET, O. YILMAZ
become
A =
0 1 0 0
2k
m
0
k
m
0
0 0 0 1
k
2m
0
k
2m
0
, B =
1
2
1
4
0 0
1
4
1
2
0 0
0 0
1
2
0
0 0 0
1
2
,
and we can evaluate that
(I −Bi)−1 =
8
3
4
3
0 0
4
3
8
3
0 0
0 0 2 0
0 0 0 2
.
The system (9) satisfies the conditions of Theorem 1, so there exists a solution which satisfies
z ≥ 0, −z′ ≥ 0 and −∆z ≥ 0, ∆z′ ≥ 0.
By retaining the original variables,(
x
y
)
≥ 0, −
(
x′
y′
)
≥ 0 and −
(
∆x
∆y
)
≥ 0,
(
∆x′
∆y′
)
≥ 0.
Example 2. Consider the following second order system:
y′′ + y′ − (sin2 t)x = 0, t 6= θi,
∆y|t=θi
= −2y + 0, 001y′, (10)
∆y′|t=θi
= 0, 2y − 3y′,
where θi = i, i = 1, 2, . . . . If the above system is identified with (3), (4), the matrix B becomes
B =
[
−2 0, 001
0, 2 −3
]
,
and one can evaluate that
(I −B)−1 =
1
3
25
30
· 10−5
1
60
1
4
,
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
POSITIVE SOLUTIONS OF LINEAR IMPULSIVE DIFFERENTIAL EQUATIONS 297
and a0 = 1, a2 = sin2 t ≥ 0. The system (10) satisfies the conditions of Theorem 2, so, there is
at least one solution y = y(t) which satisfies
y ≥ 0, −y′ ≥ 0,
−∆y ≥ 0, ∆y′ ≥ 0.
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Received 13.02.2005,
after revision — 23.05.2005
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
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