Existence and asymptotic behavior of the positive solutions of neutral impulsive differential equations
In this article we consider first order neutral impulsive differential equations with constant coefficients and
 constant delays. We study the asymptotic behavior of the eventually positive solutions of these equations
 and establish necessary and sufficient conditions for the existe...
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2005
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| Цитувати: | Existence and asymptotic behavior of the positive solutions of neutral impulsive differential equations / M.B. Dimitrova, V.I. Donev // Нелінійні коливання. — 2005. — Т. 8, № 3. — С. 304-318. — Бібліогр.: 22 назв. — англ. |
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| author | Dimitrova, M.B. Donev, V.I. |
| author_facet | Dimitrova, M.B. Donev, V.I. |
| citation_txt | Existence and asymptotic behavior of the positive solutions of neutral impulsive differential equations / M.B. Dimitrova, V.I. Donev // Нелінійні коливання. — 2005. — Т. 8, № 3. — С. 304-318. — Бібліогр.: 22 назв. — англ. |
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| description | In this article we consider first order neutral impulsive differential equations with constant coefficients and
constant delays. We study the asymptotic behavior of the eventually positive solutions of these equations
and establish necessary and sufficient conditions for the existence of such solutions.
Розглянуто нейтральнi диференцiальнi рiвняння першого порядку з iмпульсною дiєю, сталими коефiцiєнтами та сталими запiзненнями. Вивчено асимптотичну поведiнку евентуально
додатних розв’язкiв цих рiвнянь та знайдено достатнi умови їх iснування.
|
| first_indexed | 2025-12-07T17:40:07Z |
| format | Article |
| fulltext |
UDC 517 . 9
EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE POSITIVE
SOLUTIONS OF NEUTRAL IMPULSIVE DIFFERENTIAL EQUATIONS
IСНУВАННЯ ТА АСИМПТОТИЧНА ПОВЕДIНКА
ДОДАТНИХ РОЗВ’ЯЗКIВ
НЕЙТРАЛЬНИХ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
З IМПУЛЬСНОЮ ДIЄЮ
M. B. Dimitrova, V. I. Donev
Tech. Univ. Sliven
8800 Sliven, Bulgaria
In this article we consider first order neutral impulsive differential equations with constant coefficients and
constant delays. We study the asymptotic behavior of the eventually positive solutions of these equations
and establish necessary and sufficient conditions for the existence of such solutions.
Розглянуто нейтральнi диференцiальнi рiвняння першого порядку з iмпульсною дiєю, стали-
ми коефiцiєнтами та сталими запiзненнями. Вивчено асимптотичну поведiнку евентуально
додатних розв’язкiв цих рiвнянь та знайдено достатнi умови їх iснування.
1. Introduction. Impulsive differential equations with deviating arguments (IDEDA) are adequ-
ate mathematical models for simulating processes that depend on their history and are subject
to short-time disturbances. Such processes occur in the theory of optimal control, theoretical
physics, population dynamics, biotechnology, industrial robotics, etc. In contrast to the theory of
impulsive differential equations (see [1 – 4] ) and differential equations with deviating arguments
(see [5 – 9]), the theory of IDEDA, due to theoretical and practical difficulties, is developi-
ng rather slowly. We note here that [10] is the first work where IDEDA were considered. For
more results, concerning IDEDA, we choose to refer to [11 – 16]. Much less we know about the
neutral impulsive differential equations, i.e., the equations in which the highest-order derivati-
ve of the unknown function appears in the equation with the argument t (the present state of
the system), as well as with one or more retarded and/or advanced arguments (the past and/or
the future state of the system). Note that equations of this type appear in networks, containing
lossless transmission lines. Such networks arise , for example, in high speed computers, where
lossless transmission lines are used to interconnect switching circuits (see [17, 18]).
As it is known (see [5]), the appearance of the neutral term in a differential equation can
cause or destroy properties of its solutions. Moreover, the study of neutral differential equati-
ons, in general, presents complications which are unfamiliar for nonneutral differential equati-
ons. As for a discussion on some more applications and some drastic differences in behavior of
the solution of neutral differential equations see, for example, [19 – 21].
2. Preliminaries. Consider the first order neutral impulsive linear differential equation of
the form
c© M. B. Dimitrova, V. I. Donev, 2005
304 ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE POSITIVE SOLUTIONS . . . 305
[y(t)− cy(t− h)]′ + ry(t) + qy(t− σ) = 0, t 6= τk, k ∈ N,
(1)
∆[y(τk)− cy(τk − h)] + py(τk − σ) = 0, k ∈ N,
where h, σ ∈ (0,+∞), c ∈ (0, 1) and r, q, p are constants, while τk, k ∈ N , with t0 = τ0 <
< τ1 < τ2 < . . . < τk < . . . and lim
k→+∞
τk = +∞ are fixed moments of impulsive effect with
the property max {τk+1 − τk} < +∞, k ∈ N.
Denote by i(τ0, t) the number of impulses τk with τk ∈ (τ0, t), k ∈ N, and by AτC(R,R)
the set of all functions u : R → R, which are absolutely continuous in every interval (τk, τk+1),
k ∈ N. The functions u are continuous from the left as t → τk − 0, i.e., u(τk − 0) = u(τk),
k ∈ N, and may have discontinuity of the first kind at the points τk ∈ R+, k ∈ N.
Let ρ = max{σ, h}. We will say that y(t) is a solution of Eq. (1), if there exists a number
T0 ∈ R such that y ∈ AτC([T0−ρ,+∞], R), the function z(t) = y(t)−cy(t−h) is continuously
differentiable for t ≥ T0, t 6= τk, k ∈ N, and y(t) satisfies Eq. (1) for all t ≥ T0.
In what follows we will assume that every solution y(t) of the equation (1) is regular, i.e.,
y(t) is defined on [Ty,+∞) for some Ty ≥ T0 and sup {|y(t)| : t ≥ T} > 0 for each T ≥ Ty.
A regular solution y(t) of Eq. (1) is said to be eventually positive (eventually negative), if
there exists a number T1 > 0 such that y(t) > 0 (y(t) < 0) for every t ≥ T1. Also, note that
a regular solution y(t) of Eq. (1) is called nonoscillatory, if there exists a number t0 ≥ 0 such
that y(t) is of constant sign for every t ≥ t0. Otherwise, it is called oscillatory.
In this article we will study the asymptotic behavior of the eventually positive solutions
of Eq. (1) and we will establish necessary and sufficient conditions for the existence of such
solutions only in the case where Eq. (1) is a neutral impulsive differential equation, that is,
when Eq. (1) is neutral (h 6= 0 for c 6= 0) and impulsive (p 6= 0 and τk+1 − τk = h, k ∈ N).
So, in the sequel we will assume that
c ∈ (0, 1), p 6= 0, τk+1 − τk = h, k ∈ N and i(t− σ, t) = K = const. (H)
In order to obtain our results, we use an adaption of a special case of Lemma 3 of [6] (see
also [22]), which describes the asymptotic behavior of the function z(t) = y(t)− cy(t− h) and
the asymptotic behavior of the eventually positive (resp. eventually negative) solutions y(t) of
Eq. (1).
Lemma 1. Let y(t) be an eventually positive solution of Eq. (1), where r, p, q, h, σ ∈ (0,+∞)
and c ∈ (0, 1). Then:
(a) z(t) > 0 for all large t with lim
t→+∞
z(t) = 0 and lim
τk→+∞
|∆z(τk)| = 0;
(b) lim
t→+∞
y(t) = 0 and lim
τk→+∞
|∆y(τk)| = 0.
Proof. (a) Since both y(t) and −y(t) are solutions to Eq. (1), in the sequel we will consider
only eventually positive solutions y(t) of Eq. (1). That is, in what follows we will assume that
there is a number t0 > 0 such that the solution y(t) of Eq. (1) is defined on the interval [t0,+∞)
and y(t) > 0 for every t ≥ t0.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
306 M. B. DIMITROVA, V. I. DONEV
Set T0 = t0 + ρ, where ρ = max{σ, h}. Then for every t ≥ T0 and τk ≥ T0, k ∈ N, we have
[z(t)]′ < 0, t 6= τk, k ∈ N,
(2)
∆[z(τk)] < 0, k ∈ N.
Observe that, in view of (2), it follows that z(t) is a strictly decreasing function, and so lim
t→+∞
z(t)
exists in the set R ∪ {−∞}. Therefore, we have exactly one of the following three cases:
(i) lim
t→+∞
z(t) = −∞, or lim
t→+∞
z(t) = L, where L < 0 is a constant,
(ii) lim
t→+∞
z(t) = L, where L > 0 is a constant, or
(iii) lim
t→+∞
z(t) = 0.
Assume that (i) holds. Then there exists a δ > 0 and a t1 ≥ T0 such that z(t) < −δ for
every t ≥ t1, t 6= τk, k ∈ N, that is,
y(t)− cy(t− h) < −δ, t ≥ t1, t 6= τk, k ∈ N. (3)
This implies that there is an index ν, ν ∈ N, and δν > 0, such that z(τk) < −δν for every
τk ≥ τν ≥ t1, k ≥ ν.
Since ∆z(τk) < 0, τk ≥ τν ≥ t1, k ≥ ν, we see that z(τk + 0) < z(τk) < −δν for every
τk ≥ τν ≥ t1, k ≥ ν. Hence, we derive y(τk + 0)− cy(τk + 0− h) < −δν , τk ≥ τν ≥ t1, k ≥ ν
or equivalently y(τk + 0) < cy(τk + 0− h)− δν , τk ≥ τν ≥ t1, k ≥ ν and finally
y(τk + 0) < cy(τk + 0− h), τk ≥ τν ≥ t1, k ≥ ν. (4)
In view of (4), we conclude that {y(τk + 0)}, k ∈ N, for τk ≥ τν ≥ t1, k ≥ ν, ν ∈ N, is
a decreasing sequence of positive real numbers and so from (3), by consecutive iterations, we
find that
y(t) < −δ + cy(t− h) < −δ + c[−δ + cy(t− 2h)] < . . .
. . . < −δ(1 + c + c2 + . . . + cn) + cny(t− nh), n = 1, 2, . . . ,
for every t = t1 + nh, n ∈ N. Hence, we see that n =
t− t1
h
and n → +∞ as t → +∞. Since
c ∈ (0, 1), we conclude that in the last expression cn → 0, y(t − nh) = y(t1) > 0 and, since
+∞∑
k=0
ck =
1
1− c
, it follows that
lim
t→+∞
y(t) < − 1
1− c
δ, t > t1. (5)
Since {y(τk + 0)}, k ∈ N, for τk ≥ τν ≥ t1, k ≥ ν, ν ∈ N, is a decreasing sequence of
positive numbers and since y(τk) < cy(τk + 0 − h) − δν , k ∈ N, for τk ≥ τν ≥ t1, k ≥ ν,
ν ∈ N, a similar to (5) result follows for all large enough τk, k ∈ N, k ≥ ν, ν ∈ N. Namely, for
Mτ = supτk>t1{y(τk + 0)}, we obtain
y(τk + 0) < − 1
1− c
δν + cny(τk + 0− nh), k ∈ N, for τk ≥ τν ≥ t1, k ≥ ν, ν ∈ N,
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EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE POSITIVE SOLUTIONS . . . 307
or equivalently
y(τk + 0) < − 1
1− c
δν + cnMτ , k ∈ N, for τk ≥ τν ≥ t1, k ≥ ν, ν ∈ N,
and finally
y(τk + 0) < − 1
1− c
δν , k ∈ N, for τk ≥ τν ≥ t1, k ≥ ν, ν ∈ N. (6)
Now, in view of (5) and (6), we see that y(t) < − 1
1− c
max{δ, δν} < 0, ν ∈ N , for all large t.
But this contradicts the positivity of the solution y(t) and shows that the case (i) is impossible.
Consider now the case (ii). In this case, integrating Eq. (1) from T0 to t, t ≥ T0, we find that
t∫
T0
z′(s)ds +
t∫
T0
[ry(s) + qy(s− σ)]ds = 0
and hence
z(t)− z(T0)−
∑
T0≤τk<t
∆z(τk) +
t∫
T0
[ry(s) + qy(s− σ)]ds = 0.
Since ∆z(τk) = −py(τk − σ), τk ≥ T0, k ∈ N, we obtain
z(t) = z(T0)−
∑
T0≤τk<t
py(τk − σ)−
t∫
T0
[ry(s) + qy(s− σ)]ds. (7)
Because of the fact that z(t) = y(t) − cy(t − h) > L > 0 for all large t, we conclude that
0 < L < z(t) < y(t) for all large t. Using this result in (7), we obtain
z(t) ≤ z(T0)− L
∑
T0≤τk<t
p +
t∫
T0
[r + q]ds
,
which implies lim
t→+∞
z(t) = −∞ and, as in the case (i), leads to a contradiction. Thus, the case
(ii) is also impossible.
From the above observation, we see that the only possible case is that of (iii), i.e., the case,
where lim
t→+∞
z(t) = 0. Note that, since z(t), t ≥ T0, is a decreasing function, it follows that
z(t) > 0 for all large t and lim
k→+∞
|∆z(τk)| = 0, k ∈ N. This completes the proof of (a).
To prove (b), remark that, by (a), we have lim
t→+∞
[y(t)−cy(t−h)] = 0, where y(t)−cy(t−h) >
> 0 for all large t. Furthermore, lim
τk→+∞
|∆z(τk)| = 0 implies lim
τk→+∞
|∆y(τk)−c∆y(τk−h)| = 0.
Next, as in (a), we prove (2), by which z′(t) < 0 for every t ≥ T0 and hence y′(t) < cy′(t−h)
for every t ≥ T0. If we assume that y′(t) > 0 for every t ≥ T0, then obviously y′(t) < y′(t− h)
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
308 M. B. DIMITROVA, V. I. DONEV
for every t ≥ T0 and so y′ is a strictly decreasing function on the interval [T0,+∞). This implies
that y′′(t) < 0 for every t ≥ T0, and y(t) has to be increasing and concave on [T0,+∞). But
this is impossible, because y(t) is an eventually positive function and lim
t→+∞
[y(t)− cy(t− h)] =
= 0. Therefore, we see that it must be y′(t) < 0 for every t ≥ T0. This means that y(t) is a
positive decreasing piecewise continuous function in every interval of the form (τk, τk+1] for
every τk ≥ T0, k ∈ N, and changes its values at every moment τk, k ∈ N , of impulsive effect,
where lim
τk→+∞
|∆y(τk)− c∆y(τk − h)| = 0. It is clear now that lim
t→+∞
y(t) exists in R. Moreover,
since y(t)− cy(t− h) > 0 for all large t and since y(t) is decreasing as t → +∞, it follows that
lim
t→+∞
y(t) is also a finite number.
Thus, in view of the above observation, we conclude that
lim inf
t→+∞
y(t)− c lim inf
t→+∞
y(t− h) ≤ lim inf
t→+∞
[y(t)− cy(t− h)] = 0
and hence
(1− c) lim inf
t→+∞
y(t) ≤ 0,
which leads to lim inf
t→+∞
y(t) = 0 and finally to the conclusion that
lim inf
t→+∞
y(t) = lim sup
t→+∞
y(t) = lim
t→+∞
y(t) = 0
and lim
τk→+∞
|∆y(τk)| = 0, which completes the proof of (b).
The proof of the lemma is complete.
3. Main results. Analyzing the forms of the solutions of the equation (1), when c = 0 or
h = 0, i.e., when Eq. (1) is a nonneutral equation, we assume that the solutions of the neutral
impulsive differential equation (1) must be in an appropriate exponential impulsive form. So,
we will say that a function u(t) has the form of impulsive exponent, if there are real numbers A
and λ such that u(t) can be expressed in the form
u(t) = e−λtAi(τ0,t), (8)
where i(τ0, t) is the number of impulses τk with τk ∈ (τ0, t), k ∈ N. In the following the
constant A will be referred as ”pulsatile constant” of the impulsive exponent.
Remark 1. Note that, in the above formula (8), i(τ0, t) = −1 for every t ∈ (τ0 − h, τ0).
Consider Eq. (1) when r = 0 and the hypothesis (H) holds. That is, consider the equation
[y(t)− cy(t− h)]′ + qy(t− σ) = 0, t 6= τk, k ∈ N,
∆[y(τk)− cy(τk − h)] + py(τk − σ) = 0, k ∈ N,
(11)
subject to the hypothesis (H).
Our first result provides a necessary and sufficient condition for existence of an eventually
positive solution to Eq. (11) in the mentioned above form (8) of impulsive exponent.
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EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE POSITIVE SOLUTIONS . . . 309
Theorem 1. Eq. (11 ) admits an eventually positive solution y(t) in the form (8) if and only if
the algebraic equation
−λ
(
1− p
q
λ
)K
+ cλeλ,h
(
1− p
q
λ
)K−1
+ qeλσ = 0 (9)
has at least one real root λ with λ <
q
p
for pq > 0 and λ >
q
p
for pq < 0.
Proof. Let y(t) be an eventually positive solution to Eq. (11) defined by the formula (8).
Substituting y(t) into Eq. (11), from the differential equation we obtain
−λe−λtAi(τ0,t) + cλeλhAi(τ0,t−h)e−λt + qeλσAi(τ0,t−σ)e−λt = 0,
while from the impulsive conditions we have
(A− 1)Ai(τ0,τk)e−λτk − c(A− 1)eλhAi(τ0,τk−h)e−λτk + peλσAi(τ0,τk−σ)e−λτk = 0.
Having in mind the hypothesis (H), we find
i(t− σ, t) = K and i(τ0, t) = n, when t ∈ (τ0 + nh, τ0 + (n + 1)h), n ∈ N,
which leads to the system
−λAK + cλeλhAK−1 + qeλσ = 0,
(A− 1)AK − c(A− 1)eλhAK−1 + peλσ = 0.
Since for A = 1 − p
q
λ the second equation of this system coincides with the first one, it is
enough to consider the system
A = 1− p
q
λ,
λAK − cλeλhAK−1 − qeλσ = 0,
which is equivalent to the algebraic equation (9). In the sequel Eq. (9) will be referred to as
the characteristic equation of Eq. (11). Obviously, if y(t) is an eventually positive solution of
Eq. (11) in the form (8) with the positive ”pulsatile constant” A = 1 − p
q
λ, where λ <
q
p
if pq > 0 and λ >
q
p
if pq < 0, then λ satisfies the characteristic equation (9). Conversely,
consider the characteristic equation (9) and assume that it has a real root λ∗ <
q
p
if pq > 0 and
λ∗ >
q
p
if pq < 0. Then Eq. (11) admits an eventually positive solution y(t) in the form (8), i.e.,
y(t) = e−λ∗tAi(τ0,t), with the positive ”pulsatile constant” A = 1− p
q
λ∗.
The proof of the theorem is complete.
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310 M. B. DIMITROVA, V. I. DONEV
Next, consider Eq. (1) when it depends on a single deviation h = σ. That is, consider the
equation
[y(t)− cy(t− h)]′ + ry(t) + qy(t− h) = 0, t 6= τk, k ∈ N,
∆[y(τk)− cy(τk − h)] + py(τk − h) = 0, k ∈ N.
(12)
The following theorem provides a necessary and sufficient condition for the existence of an
eventually positive solution for Eq. (12) in the form (8).
Theorem 2. Eq. (12 ) admits an eventually positive solution y(t) in the form (8) if and only if
the algebraic equation
(r − λ)
(
1− p(λ− r)
q + cr
)
+ (cλ + q)eλ,h = 0 (10)
has at least one real root λ with λ < r +
q + cr
p
for p(q + cr) > 0 and λ > r +
q + cr
p
for
p(q + cr) > 0.
Proof. Let y(t) be an eventually positive solution to Eq. (12) defined by the formula (8).
Substituting y(t) into Eq. (12), from the differential equation we obtain
−λe−λtAi(τ0,t) + cλeλhAi(τ0,t−h)e−λt + re−λtAi(τ0,t) + qeλhAi(τ0,t−h)e−λt = 0
and from the impulsive conditions we get
(A− 1)Ai(τ0,τk)e−λτk − c(A− 1)eλhAi(τ0,τk−h)e−λτk + peλhAi(τ0,τk−h)e−λτk = 0.
Set, for convenience, i(τ0, τk) = n, n ∈ N. Then we obtain the following system:
−λAn + cλeλhAn−1 + rAn + qeλhAn−1 = 0, n ∈ N,
(A− 1)An − c(A− 1)eλhAn−1 + peλhAn−1 = 0, n ∈ N,
which after simplifications becomes
(r − λ)A + (cλ + q)eλh = 0,
(A− 1)A + (p− c(A− 1))eλh = 0.
If we choose A = 1 − p(λ− r)
q + cr
, then it is easy to see that the second equation of the last
system coincides with the first one. So, it is enough to consider the following system:
A = 1− p(λ− r)
q + cr
,
(r − λ)A + (cλ + q)eλh = 0
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EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE POSITIVE SOLUTIONS . . . 311
which is equivalent to the algebraic equation (10). In the following, Eq. (10) will be referred to
as the characteristic equation of Eq. (12). Obviously, if y(t) is an eventually positive solution of
Eq. (12) in the form (8) with the positive ”pulsatile constant” A = 1 − p(λ− r)
q + cr
, where λ <
< r+
q + cr
p
if p(q+cr) > 0 and λ > r+
q + cr
p
if p(q+cr) < 0, then λ satisfies the characteristic
equation (10). Convesely, consider the characteristic equation (10) and assume that it has a real
root λ∗ < r+
q + cr
p
if p(q+cr) > 0 and λ∗ > r+
q + cr
p
if p(q+cr) < 0. Then Eq. (12) admits
an eventually positive solution y(t) in the form (8), i.e., in the form y(t) = e−λ∗tAi(τ0,t), with
the positive ”pulsatile constant” A = 1− p(λ∗ − r)
q + cr
. The proof of the theorem is complete.
Consider Eq. (12) when r = 0. That is, consider the equation
[y(t)− cy(t− h)]′ + qy(t− h) = 0, t 6= τk, k ∈ N,
∆[y(τk)− cy(τk − h)] + py(τk − h) = 0, k ∈ N,
(13)
which is a special case of Eq. (11).
In the case of Eq. (13), as an immediate consequence of the last theorem, we have the
following.
Corollary 1. Equation (13 ), admits an eventually positive solution y(t) in the form (8) if and
only if the algebraic equation
−λ
(
1− p
q
λ
)
+ (cλ + q)eλ,h = 0 (11)
has at least one real root λ with λ <
q
p
for pq > 0 and λ >
q
p
for pq < 0.
As an illustration of the above result, we give the following example.
Example 1. The neutral impulsive differential equation
[y(t)− 0, 14711y(t− 1)]′ + 0, 06306365y(t− 1) = 0, t 6= τk, k ∈ N,
∆[y(τk)− 0, 14711y(τk − 1)] + 0, 1438y(τk − 1) = 0, k ∈ N,
for every t > τ0 satisfies the assumptions of Corollary 1 and, therefore, admits the eventually
positive solution y(t) = e−λ∗tA−i[τ0,t) with the initial function
ϕ(t) = Ae−λ∗t, t ∈ (τ0 − 1, τ0],
where λ∗ = 0, 1592 <
q
p
= 0, 438551 and A = 1− p
q
λ∗ = 0, 636986.
Remark 2. It is easy to see that, according to Lemma 1, this solution has the properties
lim
t→+∞
y(t) = 0, lim
τk→+∞
|∆y(τk)| = 0.
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312 M. B. DIMITROVA, V. I. DONEV
Example 2. The neutral impulsive differential equation
[y(t)− 0, 2y(t− 1)]′ + 0, 02y(t− 1) = 0, t 6= τk, k ∈ N,
∆[y(τk)− 0, 2y(τk − 1)]− 0, 03y(τk − 1) = 0, k ∈ N,
for every t > τ0 satisfies the assumptions of Corollary 1 and so admits the eventually positive
solution y(t) = e−λ∗tA−i[τ0,t) with the initial function
ϕ(t) = Ae−λ∗t, t ∈ (τ0 − 1, τ0],
where λ∗ = 0, 493 >
q
p
= −0, 666667 and A = 1− p
q
λ∗ = 0, 091.
Remark 3. Observe that this solution has the properties
lim
t→+∞
y(t) = 0, lim
τk→+∞
|∆y(τk)| = 0,
even though the assumptions of Lemma 1 are not fulfilled.
Next, we formulate and prove a corollary which provides more concrete sufficient condition
for existence of an eventually positive solution to Eq. (13) with positive coefficients q and p in
the form (8) of impulsive exponent.
Corollary 2. Equation (13 ) with positive coefficients q and p admits an eventually positive
solution in the form (8) if
1
4α2
− c
2
≤ p ≤ 1
4α1
− c
2
,
where α1 and α2 with α1 ≤ α2 are real roots of the equation
α = e
qh(c+2p)
p
α
under the assumption that qh ≤ p
(c + 2p)e
.
Proof. By Corollary 1, Eq. (13) admits an eventually positive solution y(t) if and only if its
characteristic equation (11) has a real root λ <
q
p
when pq > 0. We will try to express, in terms
of the constant coefficients and the constant delays involved in Eq. (13), how the existence of
an eventually positive solution depends on the conditions of the corollary. For this purpose
consider the function
F (λ) = −λ
(
1− p
q
λ
)
+ (cλ + q)eλh.
Clearly, F (0) = q > 0. Moreover, it is easy to check that F (λ) > 0 for every λ < 0. Indeed,
for any s > 0, set λ = −s < 0. Then, because of the fact that c ∈ (0, 1), we obtain F (−s) =
= s
(
1 +
p
q
s
)
+ (−cs + q)e−sh > s +
p
q
s2 − sce−sh + qe−sh > s(1− ce−sh) > s(1− e−sh) > 0
for every s > 0 and h > 0. Thus, we see that, if there exists a root of F (λ), it must be a positive
number, i.e., F (λ) = 0 is possible only for a number λ > 0. In order to find such a number, set
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EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE POSITIVE SOLUTIONS . . . 313
g(λ) = λ
(
1− p
q
λ
)
and f(λ) = (cλ + q)eλh and consider the graphics of the functions g(λ)
and f(λ). Remark that g(0) = g
(
q
p
)
= 0, f
(
−q
c
)
= 0 and f(0) = q. Furthermore, the
function g(λ) has its maximum gmax(λ) =
q
4p
at the point λ =
q
2p
, while the function f(λ) has
its minimum fmin(λ) = −ce−(c+qh)/h at the point λ = −c + qh
ch
. In view of this observation,
having in mind the monotonicity of the function f(λ) and the parabolic function g(λ), we will
try to locate a root λ > 0 of (11) by asking for λ =
q
2p
to satisfy the inequality
gmax
(
q
2p
)
=
q
4p
≥ f
(
q
2p
)
=
[(
cq
2p
)
+ q
]
e
qh
2p .
Hence, we conclude that
1
2(c + 2p)
≥ e
qh
2p = e
qh(c+2p)
p
1
2(c+2p) . (12)
Consider now the inequality (12) and the well-known equation
α = emα,
where m is the coefficient of proportionality.
The inequality (12) becomes an equality if and only if
qh(c + 2p)
p
=
1
e
and
1
2(c + 2p)
= e,
or equivalently if and only if qh = 2p. Therefore, Eq. (11) admits two positive real roots λ1 and
λ2, with λ2 =
q
2p
, for which F (λ1) = F (λ2) = 0. Obviously, the function y(t) = e
− q
2p
t
A−i[τ1,t),
which is of the form (8), is one of two solutions of Eq. (13).
Next, remark that the inequality (12) is satisfied if and only if α1 <
1
2(c + 2p)
< α2,
where α1 and α2 are two real roots of the equation α = e
qh(c+2p)
p
α under the assumption that
qh(c + 2p)
p
<
1
e
. Respectively Eq. (11) must have two positive real roots, say λ1 and λ2, with
λ1 <
q
2p
< λ2 and so Eq. (13) must have two eventually positive solutions of the form (8), i.e.,
of the form y(t) = e−λitA−i(τ0,t), i = 1, 2.
Generalizing the above consideration, we are led to the inequality F
(
q
2p
)
≤ 0 and conclude
that Eq. (11) admits two positive real roots λ1 and λ2 with λ1 < λ2 if and only if
α1 ≤
1
2(c + 2p)
≤ α2 ⇔
1
4α2
− c
2
≤ p ≤ 1
4α1
− c
2
,
where α1 and α2 are the roots of the equation α = e
qh(c+2p)α
p under the assumption that qh ≤
≤ p
(c + 2p)e
.
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314 M. B. DIMITROVA, V. I. DONEV
The proof of the corollary is complete.
The following example illustrates the above result.
Example 3. The neutral impulsive differential equation[
y(t)− 1
4e
y(t− 1)
]′
+
1
4e
y(t− 1) = 0, t 6= τk, k ∈ N,
∆
[
y(τk)−
1
4e
y(τk − 1)
]
+
1
8e
y(τk − 1) = 0, k ∈ N,
where τk+1 − τk = 1, k ∈ N, satisfies the assumptions of Corollary 2 and hence admits the
eventually positive solution y(t) = e−t2−i(τ0,t) for t > τ0 with the initial function
ϕ(t) = 2e−t, t ∈ (τ0 − 1, τ0].
This solution is of the form (8) and, according to Lemma 1, has the properties lim
t→+∞
y(t) = 0
and lim
τk→+∞
|∆y(τk)| = 0.
We conclude this section by considering Eq. (13) and the following neutral impulsive dif-
ferential equation
[y(t)− sy(t− h)]′ + gy(t− h) = 0, t 6= τk, k ∈ N,
(14)
∆[y(τk)− sy(τk − h)] + dy(τk − h) = 0, k ∈ N,
where g, d, h ∈ (0,+∞) and s ∈ (0, 1). The following comparison result shows that the exi-
stence of an eventually positive solution to Eq. (13) implies the existence of an eventually posi-
tive solution to Eq. (14).
Corollary 3. Consider the equations (13 ) and (14 ). If g ≤ q, d ≥ p and c+2p ≥ s+2d, then,
under the assumptions that qh ≤ p
(c + 2p)e
and (c + 2p)e
qh
2p ≤ 1
2
, the equation (14 ) admits an
eventually positive solution.
Proof. Remark that the assumptions qh ≤ p
(c + 2p)e
and (c + 2p)e
qh
2p ≤ 1
2
, according to
Corollary 2, are sufficient ones for existence of an eventually positive solution to Eq. (13).
Moreover, in view of our assumptions, it is easy to check that
1
2(s + 2d)
≥ 1
2(c + 2p)
≥ e
qh
2p ≥ e
sh
2d
and
gh ≤ qh ≤ p
(c + 2p)e
≤ d
(s + 2d)e
.
Therefore, we see that
1
2(s + 2d)
≥ e
sσ
2d
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EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE POSITIVE SOLUTIONS . . . 315
and
gh ≤ d
(s + 2d)e
.
But, in view of Corollary 2 and Theorem 2, the last two inequalities are sufficient for existence
of an eventually positive solution to Eq. (14) and the proof of the corollary is complete.
4. Applications. Since the neutral differential equation (11) in the cases where: a) p, h, σ ∈
∈ (0,+∞) and q = 0, and b) q, h, σ ∈ (0,+∞) and p = 0, has been studied in [22], in this
section we will deal with the neutral differential equation (11) in all other cases. So, Theorem 1
and its corollaries, applied to Eq. (11) in the cases under consideration give the following results.
Theorem 3. Eq. (11 ), where σ = 0, admits an eventually positive solution in form (8) if and
only if the algebraic equation
(−λ + q)
(
1− p
q
λ
)
+ cλeλh = 0
has at least one real root λ with λ <
q
p
for pq > 0 and λ >
q
p
for pq < 0.
Since this theorem is an immediate consequence of Theorem 1 and since its proof is similar
to that of Theorem 1, we omit the details of the proof.
Corollary 4. Suppose that the assumptions of Theorem 3 are satisfied. Then Eq. (11 ), where
p, q, h ∈ (0,+∞) and σ = 0, admits an eventually positive solution in the form (8), if
α1 ≤
(1− p)2
2c(1 + p)
≤ α2,
where α1 ≤ α2 are real roots of the equation
α = e
αcqh(1+p)2
p(1−p)2 ,
under the assumption that qh ≤ p(1− p)2
c(1 + p)2
1
e
.
The proof of this proposition is similar to that of Corollary 2 and therefore is omitted.
Example 4. The neutral impulsive differential equation[
y(t)− 1
6e
y(t− 1)
]′
+
ln 2
3
y(t) = 0, t 6= τk, k ∈ N,
∆
[
y(τk)−
1
6e
y(τk − 1)
]
+
1
2
y(τk) = 0, k ∈ N,
where τk+1 − τk = 1, k ∈ N, satisfies the assumptions of Theorem 3 and thus admits the
eventually positive solutions y(t) = e−λjt
(
1 − 3
2 ln 2
λj
)−i(τ0,t)
, j = 1, 2, for t > τ0 with λ1 =
= 0, 3086189 and λ2 = 0, 3329507. Note that λ1 <
q
p
and λ2 <
q
p
, where
q
p
= 0, 462.
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316 M. B. DIMITROVA, V. I. DONEV
The next theorem allows us to obtain necessary and sufficient condition for existence of an
eventually positive solutions to Eq. (11).
Theorem 4. Consider Eq. (11 ), where p, h ∈ (0,+∞) and q = σ = 0. Assume also that
maxk∈N
{
τk−τk−n
}
≤ h < mink∈N
{
τk−τk−(n+1)
}
, n ∈ N. Then Eq. (11 ) admits an eventually
positive solution of the form (8) if and only if the algebraic equation
(A− 1 + p)Ak − c(A− 1)Ak−n = 0
has at least one positive real root A ∈ (0, 1).
Proof. Let y(t) = e−λtAi(τ0,t), where A > 0 is a real number and i(τ0, t) is the number of
impulses τk, k ∈ N, with τk ∈ [τ1, t), k ∈ N, be an eventually positive solution of Eq. (11).
Substituting y(t) into Eq. (11), we obtain[
e−λtAi(τ0,t) − ce−λ(t−h)Ai(τ0,t−h)
]′
= 0,
e−λτkAi(τ0,τk)(A− 1)− ce−λτkeλh(A− 1)Ai(τ0,τk−h) + pe−λτkAi(τ0,τk) = 0
and, hence, we derive
−λAi(τ0,t) + cλeλhAi(τ0,t−h) = 0,
(13)
(A− 1)Ai(τ0,τk) − ceλh(A− 1)Ai(τ0,τk−h) + pAi(τ0,τk) = 0.
Setting λ = 0 in (13), we see that the eventually positive solution of Eq. (11) can be defined by
the equation
(A− 1 + p)Ak − c(A− 1)Ak−n = 0. (14)
Note that, as p > 0, we must have A 6= 1. Since the assumption A > 1 leads to the contradiction
Ak < cAk−n, we conclude that Eq. (11) has an eventually positive solution of the form y(t) =
= Ai(τ0,t) if and only if Eq. (14) has a positive real root A ∈ (0, 1), which is in agreement with
Lemma 1.
The proof of the theorem is complete.
Theorem 4 implies the following corollary.
Corollary 5. Consider Eq. (11 ), where p, h ∈ (0,+∞) and q = σ = 0. Assume also that
τk+1 − τk = h, k ∈ N.
Then Eq. (1) admits an eventually positive solution of the form (8) if and only if
p ≤ 1 + c−
√
4c.
Proof. As in the proof of Theorem 4, we assume the existence of an eventually positive
solution in the form (8) with A > 0 and derive Eq. (14) which in our case becomes
(A− 1)Ak − c(A− 1)Ak−1 + pAk = 0.
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EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE POSITIVE SOLUTIONS . . . 317
Hence, after some elementary manipulations, we find
A2 + (p− 1− c)A + c = 0. (15)
Observe that the two roots
A1 =
1 + c− p +
√
(p− 1− c)2 − 4c
2
and A2 =
1 + c− p−
√
(p− 1− c)2 − 4c
2
of Eq. (15) are real numbers if and only if p ≤ 1+c−
√
4c, or p ≥ 1+c+
√
4c. Remark also that
in the case where p ≥ 1 + c +
√
4c, both of the roots A1 and A2 are negative numbers, which
contradicts the fact that A > 0, and so the corresponding to them solutions are oscillatory.
Therefore, in order to have eventually positive solutions of the form (8), at least one of the
roots A1and A2 must be positive. But this is possible if and only if p ≤ 1 + c −
√
4c. Thus, we
see that Eq. (15) admits eventually positive solutions of the form y(t) = A
i(τ0,t)
j , j = 1 and/or
j = 2 which are of the form (8), if and only if p ≤ 1 + c−
√
4c.
The proof of the corollary is complete.
We conclude this article with the following result.
Theorem 5. The differential equation (1), where h > 0, r = p = q = 0, always admits an
eventually positive solution of the form y(t) = e−λ∗t(1 − λ∗)i(τ0,t), where λ∗ is a real root of the
equation
−(1− λ) + ceλh = 0. (16)
Proof. Following the proof of Theorem 1, we arrive at the system
−λAi(τ0,t) + cλeλhAi(τ0,t−h) = 0,
(A− 1)Ai(τ0,τk) − c(A− 1)eλhAi(τ0,τk−h) = 0.
By setting A = 1 − λ, we are led to the equation (16), which always has a positive root λ∗ ∈
∈ (0, 1). Consequently, Eq. (1) under the conditions of Theorem 5, always admits an eventually
positive solution of the form y(t) = e−λ∗t(1− λ∗)i(τ0,t).
The proof of the theorem is complete.
Example 5. The neutral impulsive differential equation[
y(t)− 1
6e
y(t− 1)
]′
= 0, t 6= τk, k ∈ N,
∆
[
y(τk)−
1
6e
y(τk − 1)
]
= 0, k ∈ N,
where τk+1 − τk = 1, k ∈ N, satisfies all the assumptions of Theorem 5 and, there fore, admits
the eventually positive solution y(t) = e−λtA−i(τ0,t), t > τ0, with λ = 0, 856 and A = 0, 144.
5. Acknowledgement. The authors would like to thank Professor M. K. Grammatikopulos
for his useful remarks and suggestions concerning this article.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
318 M. B. DIMITROVA, V. I. DONEV
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Received 12.05.2005
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 3
|
| id | nasplib_isofts_kiev_ua-123456789-178012 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-12-07T17:40:07Z |
| publishDate | 2005 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Dimitrova, M.B. Donev, V.I. 2021-02-17T15:51:25Z 2021-02-17T15:51:25Z 2005 Existence and asymptotic behavior of the positive solutions of neutral impulsive differential equations / M.B. Dimitrova, V.I. Donev // Нелінійні коливання. — 2005. — Т. 8, № 3. — С. 304-318. — Бібліогр.: 22 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/178012 517.9 In this article we consider first order neutral impulsive differential equations with constant coefficients and
 constant delays. We study the asymptotic behavior of the eventually positive solutions of these equations
 and establish necessary and sufficient conditions for the existence of such solutions. Розглянуто нейтральнi диференцiальнi рiвняння першого порядку з iмпульсною дiєю, сталими коефiцiєнтами та сталими запiзненнями. Вивчено асимптотичну поведiнку евентуально
 додатних розв’язкiв цих рiвнянь та знайдено достатнi умови їх iснування. en Інститут математики НАН України Нелінійні коливання Existence and asymptotic behavior of the positive solutions of neutral impulsive differential equations Існування та асимптотична поведінка додатних розв'язків нейтральних диференціальних рівнянь з імпульсною дією Существование и асимптотическое поведение положительных решений нейтральных дифференциальных уравнений с импульсным воздействием Article published earlier |
| spellingShingle | Existence and asymptotic behavior of the positive solutions of neutral impulsive differential equations Dimitrova, M.B. Donev, V.I. |
| title | Existence and asymptotic behavior of the positive solutions of neutral impulsive differential equations |
| title_alt | Існування та асимптотична поведінка додатних розв'язків нейтральних диференціальних рівнянь з імпульсною дією Существование и асимптотическое поведение положительных решений нейтральных дифференциальных уравнений с импульсным воздействием |
| title_full | Existence and asymptotic behavior of the positive solutions of neutral impulsive differential equations |
| title_fullStr | Existence and asymptotic behavior of the positive solutions of neutral impulsive differential equations |
| title_full_unstemmed | Existence and asymptotic behavior of the positive solutions of neutral impulsive differential equations |
| title_short | Existence and asymptotic behavior of the positive solutions of neutral impulsive differential equations |
| title_sort | existence and asymptotic behavior of the positive solutions of neutral impulsive differential equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/178012 |
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