Sufficient conditions for oscillation of solutions of first order neutral delay impulsive differential equations with constant coefficients
This paper is dealing with the oscillatory properties of first order neutral delay impulsive differential equations and the corresponding inequalities with constant coefficients. The established sufficient conditions ensure oscillation of every solution of this type of the equations.
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irk-123456789-1746332021-01-27T01:26:14Z Sufficient conditions for oscillation of solutions of first order neutral delay impulsive differential equations with constant coefficients Dimitrova, M.B. Donev, V.I. This paper is dealing with the oscillatory properties of first order neutral delay impulsive differential equations and the corresponding inequalities with constant coefficients. The established sufficient conditions ensure oscillation of every solution of this type of the equations. Розглянуто осциляцiйнi властивостi розв’язкiв диференцiальних рiвнянь першого порядку зi сталими коефiцiєнтами та нейтральним запiзненням iмпульсної дiї та вiдповiдних до них нерiвностей. Знайденi умови є достатнiми для того, щоб кожний розв’язок рiвняння такого типу був осциляцiйним. 2010 Article Sufficient conditions for oscillation of solutions of first order neutral delay impulsive differential equations with constant coefficients / M.B. Dimitrova, V.I. Donev // Нелінійні коливання. — 2010. — Т. 13, № 1. — С. 15-29. — Бібліогр.: 14 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/174633 517.9 en Нелінійні коливання Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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This paper is dealing with the oscillatory properties of first order neutral delay impulsive differential equations and the corresponding inequalities with constant coefficients. The established sufficient conditions
ensure oscillation of every solution of this type of the equations. |
| format |
Article |
| author |
Dimitrova, M.B. Donev, V.I. |
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Dimitrova, M.B. Donev, V.I. Sufficient conditions for oscillation of solutions of first order neutral delay impulsive differential equations with constant coefficients Нелінійні коливання |
| author_facet |
Dimitrova, M.B. Donev, V.I. |
| author_sort |
Dimitrova, M.B. |
| title |
Sufficient conditions for oscillation of solutions of first order neutral delay impulsive differential equations with constant coefficients |
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Sufficient conditions for oscillation of solutions of first order neutral delay impulsive differential equations with constant coefficients |
| title_full |
Sufficient conditions for oscillation of solutions of first order neutral delay impulsive differential equations with constant coefficients |
| title_fullStr |
Sufficient conditions for oscillation of solutions of first order neutral delay impulsive differential equations with constant coefficients |
| title_full_unstemmed |
Sufficient conditions for oscillation of solutions of first order neutral delay impulsive differential equations with constant coefficients |
| title_sort |
sufficient conditions for oscillation of solutions of first order neutral delay impulsive differential equations with constant coefficients |
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Інститут математики НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/174633 |
| citation_txt |
Sufficient conditions for oscillation of solutions of first order neutral delay impulsive differential equations with constant coefficients / M.B. Dimitrova, V.I. Donev // Нелінійні коливання. — 2010. — Т. 13, № 1. — С. 15-29. — Бібліогр.: 14 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT dimitrovamb sufficientconditionsforoscillationofsolutionsoffirstorderneutraldelayimpulsivedifferentialequationswithconstantcoefficients AT donevvi sufficientconditionsforoscillationofsolutionsoffirstorderneutraldelayimpulsivedifferentialequationswithconstantcoefficients |
| first_indexed |
2025-07-15T11:39:48Z |
| last_indexed |
2025-07-15T11:39:48Z |
| _version_ |
1837712895630114816 |
| fulltext |
UDC 517 . 9
SUFFICIENT CONDITIONS FOR OSCILLATION
OF SOLUTIONS OF FIRST ORDER NEUTRAL DELAY
IMPULSIVE DIFFERENTIAL EQUATIONS
WITH CONSTANT COEFFICIENTS
ДОСТАТНI УМОВИ ДЛЯ ОСЦИЛЯЦIЙНОСТI РОЗВ’ЯЗКIВ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ ПЕРШОГО ПОРЯДКУ
ЗI СТАЛИМИ КОЕФIЦIЄНТАМИ
ТА НЕЙТРАЛЬНИМ ЗАПIЗНЕННЯМ IМПУЛЬСНОЇ ДIЇ
M. B. Dimitrova, V. I. Donev
Techn. Univ. Sliven
8800 Sliven, Bulgaria
e-mail: vanyodi@yahoo.com
This paper is dealing with the oscillatory properties of first order neutral delay impulsive differential equati-
ons and the corresponding inequalities with constant coefficients. The established sufficient conditions
ensure oscillation of every solution of this type of the equations.
Розглянуто осциляцiйнi властивостi розв’язкiв диференцiальних рiвнянь першого порядку зi
сталими коефiцiєнтами та нейтральним запiзненням iмпульсної дiї та вiдповiдних до них нерiв-
ностей. Знайденi умови є достатнiми для того, щоб кожний розв’язок рiвняння такого типу
був осциляцiйним.
1. Introduction. In recent years, impulsive differential equations with deviating arguments
(IDEDA) attract the attention of many mathematicians. Generally speaking, IDEDA are very
interesting mixture of impulsive differential equations (see [1] and [2]) and differential equati-
ons with deviating argument (see [3 – 6]). The impulsive part of IDEDA reflects the discontinui-
ties of first kind and can be used to model mathematically the short-time disturbances of some
processes in the nature, whereas the differential part with its deviating argument (retarded,
neutral, or advanced) is a very convenient instrument for a simulation of the dependance of the
same processes on their history. Such processes occur in the theory of optimal control, theoreti-
cal physics, population dynamics, biotechnology, industrial robotics, etc. Among numerous pa-
pers published on IDEDA with retarded or advanced arguments, we choose to refer to [7 – 11].
Much less we know about the neutral impulsive differential equations (see [12 – 14]), i.e.,
the equations in which the highest-order derivative 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).
In [14], the authors obtained some criteria for oscillation of the solutions of the neutral
impulsive differential equation of first order (E1), where the coefficient c in the neutral term
belonged to the interval (0, 1). In the present paper we add and extend the results therein,
allowing for the coefficient c to be an arbitrary positive real number.
2. Preliminaries. Consider the first order neutral delay impulsive differential equation of the
c© M. B. Dimitrova, V. I. Donev, 2010
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 1 15
16 M. B. DIMITROVA, V. I. DONEV
form
d
dt
[y(t)− cy(t− h)] + py(t− σ) = 0, t 6= τk, k ∈ N, (E1)
∆ [y(τk)− cy(τk − h)] + pky(τk − σ) = 0, k ∈ N,
as well as the corresponding inequalities,
d
dt
[y(t)− cy(t− h)] + py(t− σ) ≤ 0, t 6= τk, k ∈ N, (N1,≤)
∆ [y(τk)− cy(τk − h)] + pky(τk − σ) ≤ 0, k ∈ N,
and
d
dt
[y(t)− cy(t− h)] + py(t− σ) ≥ 0, t 6= τk, k ∈ N, (N1,≥)
∆ [y(τk)− cy(τk − h)] + pky(τk − σ) ≥ 0, k ∈ N,
where the deviations h and σ and the coefficients c, p and pk are positive constants. The points
τk ∈ (0,+∞), k ∈ N, are fixed moments of the impulsive effect (let us call them jump points),
where the unknown function reveals its first kind discontinuities as jumps. Here, in order to
manifest the jumps of the unknown function y(t) in such points of the impulsive effect, we use
the notation
∆ [y(τk)− cky(τk − h)] = ∆y(τk)− ck∆y(τk − h), ∆y(τk) = y(τk + 0)− y(τk − 0).
Denote by PτC(R,R) the set of all piecewise continuous on (τk, τk+1], k ∈ N, functions
u : R → R, which at the points τk are continuous from the left, i.e., u(τk − 0) = lim
t→τk−0
u(t) =
= u(τk), for which there exists a sequence of reals {u(τk +0)} such that u(τk +0) = lim
t→τk+0
u(t)
and, which at the jump points τk, k ∈ N, may have discontinuities of the first kind which
we characterize as down-jumps if ∆u(τk) = u(τk + 0) − u(τk − 0) < 0, or as up-jumps if
∆u(τk) = u(τk + 0)− u(τk − 0) > 0, k ∈ N.
Introduce the following hypotheses:
(H1) 0 < τ1 < τ2 < . . . < τk < . . . , lim
k→+∞
τk = +∞, max {τk+1 − τk} < +∞, k ∈ N ;
(H2) p > 0, pk ≥ 0, k ∈ N, c > 0.
We will say that a function y(t) is a solution of equation (E1), if there exists a number
T0 ∈ R such that y ∈ PτC([T0,+∞], 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 equation (E1) for all t ≥ T0.
If not mentioned otherwise, we will assume throughout this paper that every solution y(t)
of equation (E1), that is under consideration here, is continuable to the right and is nontrivial.
That is, y(t) is defined on some ray of the form [Ty,+∞) and sup
{
|y(t)| : t ≥ T
}
> 0 for each
T ≥ Ty. Such a solution is called a regular solution of equation (E1).
We will say that a real-valued function u defined on an interval of the form [a,+∞) has
some property eventually, if there is a number b ≥ a such that u has this property on the
interval [b, +∞).
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 1
SUFFICIENT CONDITIONS FOR OSCILLATION OF SOLUTIONS OF FIRST ORDER NEUTRAL DELAY . . . 17
A regular solution y(t) of equation (E1) is said to be 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.
Also, note that a nonoscillatory solution is called eventually positive (eventually negative), if the
constant sign that determines its nonoscillation is positive (negative). Equation (E1) is called
oscillatory, if all its solutions are oscillatory.
Moreover, writing a functional relation (or inequality), we will mean that it holds for all
sufficiently large values of the argument.
In our investigations, we shall use two auxiliary functions composed from a solution of (E1).
So, let us consider y(t) as a solution of equation (E1) and set
z(t) = y(t)− cy(t− h), ∆z(τk) = ∆y(τk)− c∆y(τk − h), k ∈ N, (∗)
w(t) = z(t)− cz(t− h), ∆w(τk) = ∆z(τk)− c∆z(τk − h), k ∈ N. (∗∗)
We introduce, at the beginning, two lemmas which investigate the asymptotic behavior of the
functions z(t) and w(t) defined by (∗) and (∗∗), if y(t) is a non-oscillatory solution of (E1). The
first one is formulated and proved for an eventually positive solution y(t) of the equation (E1).
Lemma 1. Let y(t) be an eventually positive solution of (E1) and the hypotheses (H1), (H2)
be satisfied. Then:
(a) z(t) is an eventually decreasing function of t with down-jumps;
(b) if c ∈ (0, 1], then z(t) is an eventually positive function, i.e., z(t) > 0 for all t large enough
and lim
t→+∞
z(t) = 0 with lim
τk→+∞
|∆z(τk)| = 0;
(c) if c > 1, then z(t) is an eventually negative function, i.e., z(t) < 0 for all t large enough
and lim
t→+∞
z(t) = −∞.
Proof. (a) Let y(t) be an eventually positive solution of the equation (E1), i.e., there exists
t̃ > 0 such that y(t) is defined for every t ≥ t̃ and y(t) > 0, y(t − σ) > 0, y(t − h) > 0 for
t ≥ t̃ + max{h, σ} = t0. From (E1) and (∗) we have
z′(t) = −py(t− σ), t 6= τk, k ∈ N, t ≥ t0, (1)
∆z(τk) = −pky(τk − σ), k ∈ N, τk ≥ t0.
It follows from (1) that ∆z(τk) < 0, i.e., z(t) has "down-jumps"at the points of impulsive effect
τk and, because z′(t) < 0, we conclude that z(t) is a decreasing function for t ≥ t0. The proof
of (a) is complete.
(b) Let y(t) be an eventually positive solution of (E1). Since (a) holds, then there exists
lim
t→+∞
z(t), which can be a positive number, zero, a negative number, or −∞.
Assume that z(t) < 0 eventually. Then, because z(t) is a decreasing function with down-
jumps, for some t1 ≥ t0 there will exist δν > 0 such that z(t) < −δν for every t ≥ t1, t 6= τk,
k ∈ N, i.e.,
y(t)− cy(t− h) < −δν , t 6= τk, t ≥ t1.
Moreover, because the sequence of eventually negative numbers
{
z(τk)
}+∞
k=1
is decreasing, for
our δν > 0, there will be such a term τν in the sequence of the impulsive moments {τk},
whereafter z(τk) < −δν , for every τk ≥ τν , when k ≥ ν, k ∈ N, ν ∈ N. Hence,
y(τk)− cy(τk − h) < −δν , τk ≥ τν , k ≥ ν, k ∈ N, ν ∈ N.
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 1
18 M. B. DIMITROVA, V. I. DONEV
Denote tν = max{t1, τν} and combine the last two inequalities as
y(t) < −δν + cy(t− h), t ≥ tν ,
which is obviously also fulfilled for t = tν , too, i.e.,
y(tν) < −δν + cy(tν − h). (2)
From (2), by iterations based on t = tν + (n− 1)h, n ∈ N, we get
y(t) < −δν(1 + c + c2 + . . . + cn−1) + cny(tν − h), n ∈ N. (3)
When c = 1, it follows from (3) that y(t) becomes less than a negative number for fixed y(tν−h)
and large enough t, which is a contradiction.
If c ∈ (0, 1), the inequality (3) implies, for large enough t, that
y(t) < − δν
1− c
, t ≥ tν , (4)
and y(t) becomes less than a negative number for large enough t, which is again a contradiction.
Hence, our assumption that eventually z(t) < 0 is impossible and we conclude that eventual-
ly z(t) > 0 and lim
t→+∞
z(t) is a finite positive number or zero. The last fact implies
lim
τk→+∞
|∆z(τk)| = 0.
Further, assume lim
t→+∞
z(t) = L, L > 0. Integrating (E1) from t0 to t, we get
t∫
t0
z′(r)dr +
t∫
t0
py(r − σ)dr = 0,
or
z(t)− z(t0)−
∑
t0≤τk<t
∆z(τk) +
t∫
t0
py(r − σ)dr = 0,
or
z(t) = z(t0) +
∑
t0≤τk<t
∆z(τk)−
t∫
t0
py(r − σ)dr.
But ∆z(τk) = −pky(τk − σ) hence
z(t) = z(t0)−
∑
t0≤τk<t
pky(τk − σ)−
t∫
t0
py(r − σ)dr. (5)
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 1
SUFFICIENT CONDITIONS FOR OSCILLATION OF SOLUTIONS OF FIRST ORDER NEUTRAL DELAY . . . 19
Because z(t) = y(t)− cy(t− h), we have, in this case, that L ≤ z(t) < y(t), which determines
y(t) as a bounded function from below. Then (5) reduces to
z(t) ≤ z(t0)− L
∑
t0≤τk<t
pk +
t∫
t0
pdr
, (6)
which implies lim
t→+∞
z(t) = −∞ and contradicts our assumption. Therefore, the only possibility
left is lim
t→+∞
z(t) = 0. The proof of (b) is complete.
(c) Proceeding as in the beginning of (b), we conclude that there exists lim
t→+∞
z(t), which
can be a positive number, zero, a negative number, or −∞.
Assume lim
t→+∞
z(t) = L > 0. Then we will have L ≤ z(t) < y(t). The last fact does
determine y(t) as a bounded function from below and if we integrate (E1) from t0 to t, we can
easily get (6), which will imply lim
t→+∞
z(t) = −∞ and this will contradict our assumption.
Assume
lim
t→+∞
z(t) = lim
t→+∞
[y(t)− cy(t− h)] = L = 0. (7)
It is obvious that then z(t) > 0 eventually, i.e., there exists a number t1 ≥ t0, such that we have
y(t) > cy(t − h) > y(t − h) for every t ≥ t1. Observe that the last inequality holds as well as
for those moments of impulsive effect τk, for which τk > t1, k ∈ N. However, our assumption
implies that there will exist a strictly increasing sequence {y(tn)}∞n=1 (with moments of impulsi-
ve effect τk eventually therein), which is bounded by a positive number, i.e., lim
n→+∞
y(tn) = K,
K > 0 or which is unbounded, i.e., lim
n→+∞
y(tn) = +∞ and for which (7) has to be fulfilled. But
for this sequence we have
lim
tn→+∞
z(tn) = lim
t→∞
[y(t)−cy(t−h)] = lim
tn→+∞
y(tn)−c lim
tn→+∞
y(tn−h) = (1−c) lim
tn→+∞
y(tn) < 0
and the contradiction with (7) is evident, because c > 1.
Finally assume lim
t→+∞
z(t) = L < 0. Then, proceeding as in (b), we can get (2). Observe
that the right-hand side of (2) has to be positive, because y(t) is positive. So, we obtain the
inequality 0 < −δν + cy(t−h), which clearly shows that y(t) is a function bounded from below.
Hence, if we integrate (E1) from t0 to t, we can easily get (6), which will imply lim
t→+∞
z(t) = −∞
contradicting our assumption again.
Thus, the above considerations imply that lim
t→+∞
z(t) = −∞ and complete the proof of (c)
and of the lemma.
The second lemma is only formulated for an eventually negative solution y(t) of the equati-
on (E1), but the proof is carried out analogously to the proof of Lemma 1.
Lemma 2. Let y(t) be an eventually negative solution of (E1) and the hypotheses (H1), (H2)
be satisfied. Then:
(a) z(t) is an eventually increasing function of t with up-jumps;
(b) if c ∈ (0, 1], then z(t) is an eventually negative function, i.e., z(t) < 0 for all large enough
t and limt→+∞ z(t) = 0 with limτk→+∞ |∆z(τk)| = 0;
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 1
20 M. B. DIMITROVA, V. I. DONEV
(c) if c > 1, then z(t) is an eventually positive function, i.e., z(t) > 0 for large enough t and
limt→+∞ z(t) = +∞.
Lemmas 1, 2, applied to the functions z(t) and w(t) defined by (∗) and (∗∗), where y(t) is a
non-oscillatory solution of (E1), lead to the following useful proposition.
Lemma 3. Let y(t) be a solution of the equation (E1) and the hypotheses (H1), (H2) be
satisfied. Then:
(a) the functions z(t) and w(t) defined by (∗) and (∗∗) are also solutions of (E1);
(b) for an eventually positive function y(t), w′′(t) is an eventually positive function;
(c) for an eventually negative function y(t), w′′(t) is an eventually negative function.
Proof. (a) A direct substitution of z(t) defined by (∗) in the equation (E1) shows that z(t) is
a solution of (E1). The same holds for w(t) defined by (∗∗) as well. So, (a) is easily evident.
(b) According to (a), z(t) defined by (∗) is a solution of equation (E1). So, we have
[z(t)− cz(t− h)]′ = −[pz(t− σ)], t 6= τk.
From here, it is easy to see that
w(t)′′ = [z(t)− cz(t− h)]′′ = −p[z(t− σ)]′, t 6= τk. (8)
From (8) and Lemma 1(a) one can easily derive (b).
(c) From (8) and Lemma 2(a) one can easily derive (c).
The lemma is proved.
In order to assist our investigations on the oscillation of (E1), we consider the delay impulsi-
ve differential equation with constant coefficients of the form
z′(t) + Qz(t− s) = 0, t 6= τk, (E2)
∆z(τk) + qkz(τk − s) = 0, k ∈ N,
and the corresponding to it inequalities
z′(t) + Qz(t− s) ≤ 0, t 6= τk, (N2,≤)
∆z(τk) + qkz(τk − s) ≤ 0, k ∈ N,
and
z′(t) + Qz(t− s) ≥ 0, t 6= τk, (N2,≥)
∆z(τk) + qkz(τk − s) ≥ 0, k ∈ N,
where the deviation s and the coefficients Q, qk, k ∈ N are positive constants.
3. Oscillation of all solutions of the equation (E2). Let us note that in this section z(t) is
supposed to be self-contained and considered independently of what was set in (∗). Our aim
is to establish appropriate sufficient conditions under which the equation (E2) is oscillatory, in
order to use this result in the next section. To this end, we introduce the following hypothesis:
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 1
SUFFICIENT CONDITIONS FOR OSCILLATION OF SOLUTIONS OF FIRST ORDER NEUTRAL DELAY . . . 21
(H∗
2 ) Q > 0, 1 > qk ≥ 0, k ∈ N.
The next lemma specifies a sufficient condition for equation (E2) to be oscillatory, which
will be useful for investigation of oscillation of solutions of equation (E1).
Lemma 4. Assume that the hypotheses (H1) and (H∗
2 ) are satisfied. Suppose also that
lim sup
t→∞
∏
t−s≤τk<t
(1− qk)
< esQ, k ∈ N.
Then:
(a) the equation (E2) is oscillatory;
(b) the inequality (N2,≤) has no eventually positive solutions;
(c) the inequality (N2,≥) has no eventually negative solutions.
Proof. Since the proofs of (a), (b) and (c) can be carried out by similar arguments, it suffices
to prove only the case (a). To this end, assume the converse that equation (E2) has a non-
oscillatory solution. Since the negative of a solution of (E2) is again a solution of (E2), it suffices
to prove the lemma considering this solution as an eventually positive function.
So, suppose that there exists a solution z(t) of the equation (E2) and a number t̃ > 0 such
that z(t) is defined for t ≥ t̃ and z(t) > 0, z(t− s) > 0 for t ≥ t̃ + s = t0.
From (E2) and hypothesis (H∗
2 ), it follows that
z′(t) = −Qz(t− s) < 0, t 6= τk, k ∈ N,
∆z(τk) = −qkz(τk − s) < 0, k ∈ N,
i.e., z(t) is a positive decreasing function for t ≥ t0 with "down-jumps"at the points of impulsive
effect (∆z(τk) < 0). At the beginning we can rearrange (E2), dividing by z(t), in order to obtain
z′(t)
z(t)
= −Q
z(t− s)
z(t)
< −Q, t 6= τk, k ∈ N, (9)
∆z(τk) = −qkz(τk − s) < −qkz(τk), k ∈ N.
It follows from Condition 1, that there exists a constant L > 0 and t1 ≥ t0 such that
sQ
m
≥ L >
1
e
, t ≥ t1, (10)
where we denote
m = lim sup
t→∞
∏
t≤τk<t+s
(1− qk).
Now, integrate (9) from t− s to t, i.e.,
t∫
t−s
z(r)′
z(r)
dr < −
t∫
t−s
Qdr
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22 M. B. DIMITROVA, V. I. DONEV
and obtain
ln
z(t)
z(t− s)
+
∑
t−s≤τk<t
ln
z(τk)
z(τk + 0)
< −sQ. (11)
Also, z(τk + 0) − z(τk) = −qkz(τk − s) < −qkz(τk) and z(τk + 0) < (1 − qk)z(τk), i.e.,
1
1− qk
<
z(τk)
z(τk + 0)
. So, ln
1
1− qk
< ln
z(τk)
z(τk + 0)
and from (10) and (11) we get
ln
z(t)
z(t− s)
∏
t−s≤τk<t
1
1− qk
< −sQ,
i.e.,
ln
z(t− s)
z(t)
∏
t−s≤τk<t
(1− qk)
> Lm.
Using the inequality ex > ex, it follows from the last inequality that
z(t− s)
z(t)
∏
t−s≤τk<t
(1− qk) > eLm,
which implies
z(t− s)
z(t)
> eL.
Repeating the above procedure by induction (9), we conclude that there exists a sequence {tn},
where tn → ∞ as n → ∞, such that
z(t− s)
z(t)
> (eL)n, t ≥ tn. (12)
Choose n such that (
2
mL
)2
< (eL)n , (13)
which is possible because we have eL > 1 by (10). Further, fix arbitrary chosen t̂, where t̂ ≥ tn.
Because of (10), there exists a ξ ∈ [t̂− s, t̂] such that
(ξ − t̂ + s)Q ≥ mL
2
, (t̂− ξ)Q ≥ mL
2
.
Integrating (E2) over the interval [t̂− s, ξ], we find
z(ξ)− z(t̂− s)−
∑
t̂−s≤τi≤ξ
∆z(τi) +
ξ∫
t̂−s
Qz(r − s)dr = 0,
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SUFFICIENT CONDITIONS FOR OSCILLATION OF SOLUTIONS OF FIRST ORDER NEUTRAL DELAY . . . 23
or
z(ξ)− z(t̂− s) +
∑
t̂−s≤τi≤ξ
qiz(τi − s) + Q
ξ∫
t̂−s
z(r − s)dr = 0.
By omitting the first and the third terms and using the decreasing nature of z(t) we find
z(t̂− s) > (ξ − t̂ + s)Qz(ξ − s),
i.e.,
z(t̂− s)
z(ξ − s)
>
mL
2
. (14)
Similarly, integrating (E2) over the interval [ξ, t̂], we find
z(t̂)− z(ξ)−
∑
ξ≤τi≤t̂
∆z(τi) +
t̂∫
ξ
Qz(r − s)dr = 0,
or
z(t̂)− z(ξ) +
∑
ξ≤τi≤t̂
qiz(τi − s) +
t̂∫
ξ
Qz(r − s)dr = 0.
By omitting the first and the third terms and using the decreasing nature of z(t) we find
z(ξ) > (t̂− ξ)Qz(t̂− s),
i.e.,
z(ξ)
z(t̂− s)
>
mL
2
. (15)
From (14) and (15) we conclude
z(ξ − s)
z(ξ)
<
(
2
mL
)2
,
which, together with (12), imply
(eL)n <
z(ξ − s)
z(ξ)
<
(
2
mL
)2
. (16)
Note that (16) is in contradiction with (13).
The lemma is proved.
4. Oscillation of all solutions of equation (E1). Using the results of the previous two secti-
ons, we establish sufficient conditions for equation (E1) to be oscillatory.
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24 M. B. DIMITROVA, V. I. DONEV
Theorem 1. Assume that the hypotheses (H1), (H2) are satisfied. Suppose also that:
(a) c <
1
pk
, pk ≥ 1, k ∈ N ;
(b) lim sup
t→∞
∏
t−h−σ≤τk<t
(1− cpk)
≤ ecp(σ + h).
Then the equation (E1) is oscillatory.
Proof. Assume, conversely, that equation (E1) has a non-oscillatory solution. Since the
negative of a solution of (E1) is again a solution of (E1), it suffices to prove the theorem consi-
dering an eventually positive solution of (E1).
So, let us suppose that there exists a solution y(t) of the equation (E1) and a number t̃ > 0
such that y(t) is defined for t ≥ t̃, y(t) > 0 for t ≥ t̃ and y(t−h) > 0, y(t−σ) > 0 for t ≥ t0 =
= t̃ + max{h, σ}. From Lemma 1(a), it follows that z(t) which is defined by (∗) is a decreasing
function for t ≥ t0 with "down-jumps"at the points of impulsive effect (∆z(τk) < 0), and by
Lemma 1(b), it follows that z(t) is eventually positive.
Then, there exists some t1 ≥ t0 such that z(t) > 0 for t ≥ t1 ≥ t0 with "down-jumps"at
the points of impulsive effect (∆z(τk) < 0). From the fact that y(t) − cy(t − h) = z(t) > 0,
it is easy to conclude that y(t) > z(t), as well as y(t) > cy(t − h). Obviously, then we have
y(t− σ) > cy(t− σ − h) > cz(t− σ − h), i.e.,
y(t− σ) > cz(t− σ − h). (17)
Multiplying both sides of (17) by −p < 0 we obtain
z′(t) = −py(t− σ) < −cpz(t− σ − h).
Hence,
z′(t) + cpz(t− σ − h) < 0. (18)
Observe that from (17) we have that cz(τk − σ − h) < y(τk − σ), k ∈ N.
Multiplying by −pk < 0, k ∈ N, both sides of the last inequality, we obtain that
−cpkz(τk + h− σ) > −pky(τ − σ) = ∆z(τk), k ∈ N,
i.e.,
∆z(τk) + cpkz(τk − σ − h) < 0, k ∈ N. (19)
From (18) and (19), denoting s = σ + h > 0, Q = cp, qk = cpk, we see that the positive
function z(t) satisfies the delay impulsive differential inequality of the form
z(t)′ + Qz(t− s) < 0, t 6= τk, k ∈ N, (20)
∆z(τk) + qkz(τk − s) < 0, k ∈ N,
which contradicts Lemma 4(b).
The theorem is proved.
As an immediate consequence of the above theorem, we formulate the following result.
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SUFFICIENT CONDITIONS FOR OSCILLATION OF SOLUTIONS OF FIRST ORDER NEUTRAL DELAY . . . 25
Corollary 1. Let the conditions of Theorem 1 be satisfied. Then:
(a) the inequality (N1,≤) has no eventually positive solutions;
(b) the inequality (N1,≥) has no eventually negative solutions.
The proof of the corollary is similar to that of Theorem 1 and is omitted.
Theorem 2. Assume that the hypotheses (H1), (H2) are satisfied. Suppose also that:
(a) σ > h > 0 and c + pk < 1, k ∈ N ;
(b) lim sup
t→∞
∏
t+h−σ≤τk<t
(
1− pk
1− c
) ≤ ep(σ − h)
1− c
.
Then the equation (E1) is oscillatory.
Proof. Assume, for the sake of contradiction, that equation (E1) has an eventually positive
solution y(t). Then there exists t̃ > 0 such that y(t) > 0 for every t > t̃. Also, there is t0 ≥ t̃+σ
such that y(t − σ) > 0 and ∆[y(τk) − cy(τk − h)] = −pky(τk − σ) < 0, k ∈ N, for every
t ≥ t0. Now, by the conditions of the theorem, Lemma 3(a) and Lemma 1(b), it follows that
the functions z(t) and w(t), defined respectively by (∗) and (∗∗), will be eventually positive
decreasing solutions to the equation (E1) for every t ≥ t0. That is, w(t) satisfies, as a decreasing
positive solution, the equation
d
dt
[w(t)− cw(t− h)] + pw(t− σ) = 0, t 6= τk, (21)
∆[w(τk)− cw(τk − h)] + pkw(τk − σ) = 0, k ∈ N.
Let us recall that, by Lemma 3(b), w′(t) is an increasing function. Therefore, from (21) it is easy
to see that
w′(t− h)− cw′(t− h) + pw(t− σ) ≤
≤ w′(t)− cw′(t− h) + pw(t− σ) =
=
d
dt
[w(t)− cw(t− h)] + pw(t− σ) = 0. (22)
Moreover, since z(t) is a decreasing function, we see that z(τk − σ) < z(τk − σ − h) and so,
using the definitions of the functions z(t) and w(t), it is easy to conclude that
∆w(τk) = −pkz(τk − σ) > −pkz(τk − σ − h) = ∆w(τk − h), k ∈ N.
So, in view of the above observation, from (21) it follows that, for each k ∈ N,
∆w(τk − h)− c∆w(τk − h) + pkw(τk − σ) ≤
≤ ∆w(τk)− c∆w(τk − h) + pkw(τk − σ) =
= ∆[w(τk)− cw(τk − h)] + pkw(τk − σ) = 0. (23)
Now, from (22) and (23), it follows that w(t) is an eventually positive function for which
(1− c)w′(t− h) + pw(t− σ) ≤ 0, t 6= τk,
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26 M. B. DIMITROVA, V. I. DONEV
(1− c)∆w(τk − h) + pkw(τk − σ) ≤ 0, k ∈ N.
Divide the last two inequalities by 1 − c > 0 and denote s = σ − h, Q =
p
1− c
, qk =
pk
1− c
.
Then, substituting t̂ = t − h we conclude that w(t) becomes an eventually positive solution of
the delay impulsive differential inequality of the form
w′(t̂) + Qw(t̂− s) ≤ 0, t 6= τk, (24)
∆w(τk) + qkw(τk − s) ≤ 0, k ∈ N.
But the last conclusion, contradicts Lemma 4(b) and completes the proof.
As an immediate consequence of the above theorem, we formulate the following result.
Corollary 2. Let the conditions of Theorem 2 be satisfied. Then:
(a) the inequality (N1,≤) has no eventually positive solutions;
(b) the inequality (N1,≥) has no eventually negative solutions.
The proof of the corollary is similar to that of Theorem 2 and is omitted.
Note that the previous two theorems considered the oscillation of the equation (E1) under
the condition c ∈ (0, 1). Next we give a result in the case where c > 1.
Theorem 3. Assume that the hypotheses (H1) – (H∗
2 ) are satisfied. Suppose also that:
(a) σ ≤ h, c > 1;
(b) lim inf
t→+∞
p(h− σ) +
∑
t+σ≤τk≤t+h
pk
≥ c2.
Then equation (E1) is oscillatory.
Proof. Assume, for the sake of contradiction, that y(t) is an eventually positive solution of
the equation (E1). Then, in view of Lemma 3(a) and Lemma 1(c), the function z(t), which is
defined by (∗), will be an eventually negative decreasing solution to the equation (E1). That is,
[z(t)− cz(t− h)]′ + pz(t− σ) = 0, t 6= τk, (25)
∆[z(τk)− cz(τk − h)] + pkz(τk − σ) = 0, k ∈ N.
Integrating (25) from t + σ to t + h, we obtain
z(t + h)− z(t + σ)− c[z(t)− z(t + σ − h)] +
∑
t+σ≤τk≤t+h
pkz(τk − σ) +
t+h∫
t+σ
pz(r − σ)dr = 0.
By omitting the first term and counting the decreasing nature of z(t) we have
−z(t + σ)− cz(t) + cz(t + σ − h) +
∑
t+σ≤τk≤t+h
pkz(τk − σ) + p(h− σ)z(t) > 0,
or, because z(t) < z(t− h), we can get
−z(t + σ)− cz(t) + cz(t + σ − h) +
∑
t+σ≤τk≤t+h
pkz(τk − σ) + p(h− σ)z(t− h) > 0. (26)
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SUFFICIENT CONDITIONS FOR OSCILLATION OF SOLUTIONS OF FIRST ORDER NEUTRAL DELAY . . . 27
Observe that w(t) defined by (∗∗) will be a positive function of t, by Lemma 2(c). Therefore,
w(t) = z(t)− cz(t− h) > 0 leads us to the conclusion that z(t) > cz(t− h). Hence,
−cz(t) < −c2z(t− h) and − z(t + σ) < −cz(t + σ − h). (27)
From (26) and (27), it follows that
−cz(t+σ−h)−c2z(t−h)+cz(t+σ−h)+
∑
t+σ≤τk≤t+h
pkz(τk−σ)+p(h−σ)z(t−h) > 0. (28)
Moreover, because of the decreasing nature of z(t), we have z(τk − σ) ≤ z(t) < z(t − h) for
t + σ ≤ τk ≤ t + h and so, from (28), we derive
−c2z(t− h) + z(t− h)
∑
t+σ≤τk≤t+h
pk + p(h− σ)z(t− h) > 0,
or, finally,
z(t− h)
−c2 + p(h− σ) +
∑
t+σ≤τk≤t+h
pk
> 0,
which, because of the negative function z(t), implies
p(h− σ) +
∑
t+σ≤τk≤t+h
pk < c2.
The conclusion obtained contradicts Condition 2 of the theorem and completes the proof.
As an immediate consequences of the above theorem, we get the following results.
Theorem 4. Assume that the hypotheses (H1) – (H∗
2 ) are satisfied. Suppose also that:
(a) σ ≤ h, c > 1;
(b) 0 ≤ p(h− σ) ≤ c
e
;
(c) lim inf
t→+∞
∑
t+σ≤τk≤t+h
pk ≥ c2.
Then equation (E1) is oscillatory.
Remark 1. As is known (see, for example, [3], Corollary 3.1.6, or [5], Theorem 6.4.2), a
sufficient condition, for oscillation of the neutral delay differential equation
d
dt
[y(t)− cy(t− h)] + py(t− σ) = 0, (E∗
1)
without impulse effects, is
p
c
(h − σ) >
1
e
. However, the presence of the impulse effects in
the differential equation can cause or destroy the oscillation of its solutions. Our result above
demonstrates in partiular the influence of the appearance of the impulse effects on the behavior
of solutions of (E1). Indeed, Theorem 4 shows that the neutral delay impulsive differential
equation (E1) is oscillatory even in the case where
p
c
(h− σ) ≤ 1
e
.
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28 M. B. DIMITROVA, V. I. DONEV
Corollary 3. Let the conditions of Theorem 3 or Theorem 4 be satisfied. Then:
(a) the inequality (N1,≤) has no eventually positive solutions;
(b) the inequality (N1,≥) has no eventually negative solutions.
5. Examples. Example 1. Consider the neutral delay impulsive differential equation
[y(t)− 0.2y(t− 2)]′ +
1
10
y(t− 1) = 0, t 6= τk,
∆[y(τk)− 0.2y(τk − 2)] + y(τk − 1) = 0, k ∈ N,
where p =
1
10
, pk = 1, c = 0.2, h = 2 > σ = 1 and τk+1 − τk = 1, k ∈ N, e = exp(1).
Here, the hypotheses (H1), (H2) are satisfied and c ∈ (0, 1). It is easy to check that the
conditions of Theorem 1 are satisfied. Indeed, c = 0.2 <
1
pk
= 1, pk ≥ 1, k ∈ N, and
lim sup
t→∞
[ ∏
t−h−σ≤τk<t
(1− cpk)
]
= lim sup
t→∞
[ ∏
t−3≤τk<t
(0.8)
]
= 0.64 ≤ ecp(σ + h) = 0.6e.
Therefore, by Theorem 1, all solutions of the above equation are oscillatory. For example,
one oscillatory solution of this equation is the "pulsatile exponent"(see, [12] for the definition
of such a "pulsatile"function) y(t) = Ai[t0,t)e−λ∗t, t0 ≥ 0, where A = −0.2226870709 is the
"pulsatile constant"and λ∗ = 0.12226870709 is the exponential argument.
Example 2. Consider the neutral delay impulsive differential equation
[y(t)− 1.5y(t− 2)]′ +
1
e
y(t− 1) = 0, t 6= τk,
∆[y(τk)− 1.5y(τk − 2)] + 4y(τk − 1) = 0, k ∈ N,
where p =
1
e
, pk = 4, c = 1.5, h = 2 > σ = 1 and τk+1 − τk = 1, k ∈ N, e = exp(1).
Here, the hypotheses (H1), (H2) are satisfied and c ∈ (1,+∞). It is easy to check that the
conditions of Theorem 3, as well as the conditions of Theorem 4 are satisfied. Indeed,
lim inf
t→+∞
p(h− σ) +
∑
t+σ≤τk≤t+h
pk
= lim inf
t→+∞
1
e
(2− 1) +
∑
t+1≤τk≤t+2
4
= 4 +
1
e
≥ c2 = 2.25.
Therefore, by Theorem 3, or by Theorem 4, all solutions of the above equation are osci-
llatory. For example, one oscillatory solution of this equation is the "pulsatile exponent"y(t) =
= Ai[t0,t) e−λ∗t, t0 ≥ 0, where A = −0.5698188218 is the "pulsatile constant"and λ∗ =
= 0.14437601777 is the exponential argument.
6. Acknowledgement. The authors dedicate this work to the memory of their teacher Dr.
M. K. Grammatikopulos.
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Received 05.11.08
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