Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems
UDC 517.9 We consider the second order impulsive differential equation with delays  $$(Lx)(t)\equiv x''(t) + \sum\limits _{j = 1}^{p} a_j(t) x'(t-\tau_j(t)) + \sum\limits _{j = 1}^{p} b_j(t) x(t-\theta_j(t)) = f(t),  \quad t \in [0, \omega],&...
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Institute of Mathematics, NAS of Ukraine
2021
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507035664121856 |
|---|---|
| author | Domoshnitsky, A. Mizgireva, Iu. Raichik, V. Домошнитский, Александр Mizgireva, Iuliia Domoshnitsky, A. Mizgireva, Iu. Raichik, V. |
| author_facet | Domoshnitsky, A. Mizgireva, Iu. Raichik, V. Домошнитский, Александр Mizgireva, Iuliia Domoshnitsky, A. Mizgireva, Iu. Raichik, V. |
| author_sort | Domoshnitsky, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:47:53Z |
| description | UDC 517.9
We consider the second order impulsive differential equation with delays 
$$(Lx)(t)\equiv x''(t) + \sum\limits _{j = 1}^{p} a_j(t) x'(t-\tau_j(t)) + \sum\limits _{j = 1}^{p} b_j(t) x(t-\theta_j(t)) = f(t),  \quad t \in [0, \omega], $$ $$x(t_k) = \gamma_k x(t_k-0), \quad x'(t_k) = \delta_k x'(t_k-0),\quad  k = 1, 2, \ldots , r,$$
where $\gamma_k > 0,$ $\delta_k >0$ for $k = 1, 2, \ldots , r.$ In this paper, we obtain the conditions of semi-nonoscillation for the corresponding homogeneous equation on the interval $[0, \omega].$  Using these results, we formulate theorems on sign-constancy of Green's functions for two-point impulsive boundary-value problems in terms of differential inequalities. 
|
| doi_str_mv | 10.37863/umzh.v73i7.473 |
| first_indexed | 2026-03-24T02:02:55Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i7.473
UDC 517.9
A. Domoshnitsky, Iu. Mizgireva, V. Raichik (Ariel Univ., Israel)
SEMI-NONOSCILLATION INTERVALS
AND SIGN-CONSTANCY OF GREEN’S FUNCTIONS
OF TWO-POINT IMPULSIVE BOUNDARY-VALUE PROBLEMS*
IНТЕРВАЛИ МАЙЖЕ ВIДСУТНОСТI КОЛИВАНЬ
ТА ЗБЕРЕЖЕННЯ ЗНАКА ФУНКЦIЙ ГРIНА
ДВОТОЧКОВИХ IМПУЛЬСНИХ КРАЙОВИХ ЗАДАЧ
We consider the second order impulsive differential equation with delays
(Lx)(t) \equiv x\prime \prime (t) +
p\sum
j=1
aj(t)x
\prime (t - \tau j(t)) +
p\sum
j=1
bj(t)x(t - \theta j(t)) = f(t), t \in [0, \omega ],
x(tk) = \gamma kx(tk - 0), x\prime (tk) = \delta kx
\prime (tk - 0), k = 1, 2, . . . , r,
where \gamma k > 0, \delta k > 0 for k = 1, 2, . . . , r. In this paper, we obtain the conditions of semi-nonoscillation for the
corresponding homogeneous equation on the interval [0, \omega ]. Using these results, we formulate theorems on sign-constancy
of Green’s functions for two-point impulsive boundary-value problems in terms of differential inequalities.
Розглядається iмпульсне диференцiальне рiвняння другого порядку iз запiзненням
(Lx)(t) \equiv x\prime \prime (t) +
p\sum
j=1
aj(t)x
\prime (t - \tau j(t)) +
p\sum
j=1
bj(t)x(t - \theta j(t)) = f(t), t \in [0, \omega ],
x(tk) = \gamma kx(tk - 0), x\prime (tk) = \delta kx
\prime (tk - 0), k = 1, 2, . . . , r,
де \gamma k > 0, \delta k > 0 для k = 1, 2, . . . , r. Знайдено умови майже вiдсутностi коливань для вiдповiдного однорiдного
рiвняння на iнтервалi [0, \omega ]. За допомогою цих результатiв сформульовано теореми про збереження знака функцiй
Грiна двоточкових iмпульсних крайових задач у термiнах диференцiальних нерiвностей.
1. Introduction. Impulsive differential equations have attracted an attention of a number of re-
cognized mathematicians and have applications in many spheres of science from physics, biology,
medicine to economical studies. The following well-known books can be noted in this context
[20, 23 – 25]. In the books [4, 5], the concept of the general theory of functional differential equations
was presented. On the basis of this concept nonoscillation for the first order functional differential
equations was considered in [7], where positivity of the Cauchy and Green’s functions of the periodic
problem was firstly studied. A concept of nonoscillation for the first order differential equations
is also considered in the book [1]. Sign properties of the first order impulsive initial, one-point
and periodic boundary-value problems were studied in [13], and for nonlocal problems in [11, 12].
The positivity of solutions and sign-constancy of Green’s function of one- and two-point impulsive
boundary-value problems for second order functional differential impulsive equations was studied in
[2, 8, 9, 14 – 17, 21, 22].
Let us consider the impulsive equation
* This paper is part of the second author’s Ph.D. thesis which is being carried out in the Department of Mathematics at
Ariel University.
c\bigcirc A. DOMOSHNITSKY, IU. MIZGIREVA, V. RAICHIK, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7 887
888 A. DOMOSHNITSKY, IU. MIZGIREVA, V. RAICHIK
(Lx)(t) \equiv x\prime \prime (t) +
p\sum
j=1
aj(t)x
\prime (t - \tau j(t)) +
p\sum
j=1
bj(t)x(t - \theta j(t)) = f(t), t \in [0, \omega ], (1.1)
x(tk) = \gamma kx(tk - 0), x\prime (tk) = \delta kx
\prime (tk - 0), k = 1, 2, . . . , r,
0 = t0 < t1 < t2 < . . . < tr < tr+1 = \omega ,
(1.2)
x(\zeta ) = 0, x\prime (\zeta ) = 0, \zeta < 0, (1.3)
where f, aj , bj : [0, \omega ] \rightarrow \BbbR are summable functions and \tau j , \theta j : [0, \omega ] \rightarrow [0,+\infty ) are measurable
functions for j = 1, 2, . . . , p, p and r are natural numbers, \gamma k and \delta k are real positive numbers.
Let D(t1, t2, . . . , tr) be a space of functions x : [0, \omega ] \rightarrow \BbbR such that their derivative x\prime (t)
is absolutely continuous on every interval t \in [ti, ti+1), i = 0, 1, . . . , r, x\prime \prime \in L\infty , we assume
also that there exist the finite limits x(ti - 0) = \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow t - i
x(t) and x\prime (ti - 0) = \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow t - i
x\prime (t) and
condition (1.2) is satisfied at points ti, i = 0, 1, . . . , r. As a solution x we understand a function
x \in D(t1, t2, . . . , tr) satisfying (1.1) – (1.3).
It should be noted that the space of solutions of the homogeneous equation
(Lx)(t) \equiv x\prime \prime (t) +
p\sum
j=1
aj(t)x
\prime (t - \tau j(t)) +
p\sum
j=1
bj(t)x(t - \theta j(t)) = 0, t \in [0, \omega ], (1.4)
x(tk) = \gamma kx(tk - 0), x\prime (tk) = \delta kx
\prime (tk - 0), k = 1, 2, . . . , r,
0 = t0 < t1 < t2 < . . . < tr < tr+1 = \omega ,
(1.5)
x(\zeta ) = 0, x\prime (\zeta ) = 0, \zeta < 0, (1.6)
is two-dimensional. The fundamental system of solutions of impulsive differential equation (1.4) –
(1.6) consists of two linearly independent solutions x1 and x2. Like in the case of ordinary differential
equations its Wronskian
W (t) =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
x1(t) x2(t)
x\prime 1(t) x\prime 2(t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
could be one of the classical objects in this theory. Various tests, where the Wronskian does not
vanish for the nonimpulsive equation were obtained in [3, 6, 19].
2. Preliminaries. Wronskian is one of the classical objects in the theory of differential equations.
Properties of Wronskian lead to important conclusions on behavior of solutions of delay equations
[6]. The following theorem claims that nonvanishing Wronskian ensures validity of Sturm separation
theorem (between two adjacent zeros of any solution there is one and only one zero of every other
nontrivial linearly independent solution) for delay impulsive equation [10]
x\prime \prime (t) + b1x(t - \theta 1) = 0, (2.1)
with the conditions (1.2), (1.3).
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SEMI-NONOSCILLATION INTERVALS AND SIGN-CONSTANCY OF GREEN’S FUNCTIONS . . . 889
Theorem 2.1. Let p = 1, b1(t) \geq 0, t - \theta 1(t) be nondecreasing, \gamma i > 0 and \delta i > 0 for every
i = 1, r. If the Wronskian W (t) of the fundamental system of (1.4) – (1.6) does not have zeros, then
Sturm’s separation theorem is valid.
In contrast to this assertion, in this paper, the case aj(t) \geq 0, bj(t) \leq 0, j = 1, . . . , p, t \in [0, \omega ],
is considered and the number of terms can be p > 1. We focus on the connection between nonva-
nishing Wronskian, the property of semi-nonoscillation and the sign-constancy of Green’s functions
for impulsive boundary-value problems.
Below the following definition will be used.
Definition 2.1. We call [0, \omega ] a semi-nonoscillation interval of (1.4) – (1.6), if every nontrivial
solution having zero of derivative does not have zero on this interval.
For equation (1.1) – (1.3) we consider the following variants of boundary conditions:
x(0) = 0, x(\omega ) = 0, (2.2)
x\prime (0) = 0, x(\omega ) = 0, (2.3)
x(0) = 0, x\prime (\omega ) = 0, (2.4)
x\prime (0) = 0, x\prime (\omega ) = 0. (2.5)
General solution of the equation (1.1) – (1.3) can be represented in the form [7]
x(t) = \nu 1(t)x(0) + C(t, 0)x\prime (0) +
t\int
0
C(t, s)f(s)ds, (2.6)
where \nu 1(t) is a solution of the homogeneous equation (1.4) – (1.6) with the initial conditions x(0) =
= 1, x\prime (0) = 0; C(t, s), called the Cauchy function of the equation (1.4) – (1.6), is the solution of
the equation
(Lsx)(t) \equiv x\prime \prime (t) +
p\sum
j=1
aj(t)x
\prime (t - \tau j(t)) +
p\sum
j=1
bj(t)x(t - \theta j(t)) = 0, t \in [s, \omega ],
x(tk) = \gamma kx(tk - 0), x\prime (tk) = \delta kx
\prime (tk - 0), k = m, . . . , r,
0 = t0 < t1 < t2 < . . . < tr < tr+1 = \omega .
Here, m is a number such that tm - 1 < s \leq tm,
x(\zeta ) = 0, x\prime (\zeta ) = 0, \zeta < s,
satisfying the initial conditions C(s, s) = 0, C \prime
t(s, s) = 1 and C(t, s) = 0 for t < s.
If the boundary-value problem (1.1) – (1.3), (2.1) – (2.4) is uniquely solvable, then its solution can
be represented as
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
890 A. DOMOSHNITSKY, IU. MIZGIREVA, V. RAICHIK
x(t) =
\omega \int
0
Gi(t, s)f(s)ds, i = 1, 4,
where Gi(t, s) is Green’s function of the problem (1.1) – (1.3), (2.1) – (2.4), respectively [8].
Using general representation of the solution (2.6), the following formulas for Green’s functions
can be obtained:
G1(t, s) = C(t, s) - C(t, 0)
C(\omega , s)
C(\omega , 0)
, (2.7)
G2(t, s) = C(t, s) - C(\omega , s)
\nu 1(t)
\nu 1(\omega )
, (2.8)
G3(t, s) = C(t, s) - C(t, 0)
C \prime
t(\omega , s)
C \prime
t(\omega , 0)
, (2.9)
G4(t, s) = C(t, s) - C \prime
t(\omega , s)
\nu 1(t)
\nu \prime 1(\omega )
. (2.10)
Denote G\xi (t, s) the Green’s function of the problem (1.1) – (1.3) with boundary conditions
x(\xi ) = 0, x\prime (\xi ) = 0. (2.11)
In the paper [8], the following theorem has been proven for the problems (1.1) – (1.3), (2.1) –
(2.4).
Lemma 2.1. Assume that the following conditions are fulfilled:
(1) bj(t) \leq 0, j = 1, . . . , p, t \in [0, \omega ];
(2) the Cauchy function C1(t, s) of the first order impulsive equation
y\prime (t) +
p\sum
j=1
aj(t)y(t - \tau j(t)) = 0, t \in [0, \omega ],
y(tk) = \delta ky(tk - 0), k = 1, 2, . . . , r,
y(\zeta ) = 0, \zeta < 0,
(2.12)
is positive for 0 \leq s \leq t \leq \omega ;
(3) Green’s function G\xi (t, s) of the problem (1.1) – (1.3), (2.11) is nonnegative for t, s \in [0, \xi ]
for every 0 < \xi < \omega ;
(4) [0, \omega ] is a semi-nonoscilation interval of (Lx)(t) = 0.
Then Green’s functions Gi(t, s), i = 1, 3, are nonpositive for t, s \in [0, \omega ] and under the additional
condition
\sum p
j=1
bj(t)\chi [0,\omega ](t - \theta j(t)) \not \equiv 0, t \in [0, \omega ], where
\chi [0,\omega ](t) =
\left\{ 1, t \in [0, \omega ],
0, t \not \in [0, \omega ],
(2.13)
G4(t, s) \leq 0 for t, s \in [0, \omega ].
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
SEMI-NONOSCILLATION INTERVALS AND SIGN-CONSTANCY OF GREEN’S FUNCTIONS . . . 891
3. Conditions of semi-nonoscillation. Let us formulate a theorem on semi-nonoscillation of the
interval [0, \omega ].
Theorem 3.1. Assume that the following conditions are fulfilled:
(1) aj(t) \geq 0, bj(t) \leq 0, j = 1, . . . , p, t \in [0, \omega ];
(2) the Wronskian W (t) of the fundamental system of solutions of homogeneous equation (1.4) –
(1.6) satisfies the inequality W (t) \not = 0, t \in [0, \omega ];
(3) the Cauchy function C1(t, s) of the first order equation (2.12) is positive for 0 \leq s \leq t \leq \omega .
Then the interval [0, \omega ] is a semi-nonoscillation interval of (1.4) – (1.6).
Remark 3.1. It looks that conditions 2 and 3 are hard for verification. In Section 4, we explain
how these conditions could be verified in almost all cases which come from various applications.
The first conditions on positivity of the Cauchy function of the equation (2.12) are obtained in [7].
Proof. First of all, let us prove that if the conditions 1 and 3 of Theorem 3.1 are fulfilled, then
there exists a positive solution x(t) of the equation (1.4) – (1.6), satisfying the conditions x(0) =
= \alpha > 0, x\prime (0) > 0.
It is clear that in this case, x(t) satisfies the equation
x\prime \prime (t) +
p\sum
j=1
aj(t)x
\prime (t - \tau j(t)) = \phi (t), t \in [0, \omega ], (3.1)
where \phi (t) = -
\sum p
j=1
bj(t)x(t - \theta j(t)) for t \in [0, \omega ].
Let us denote y(t) = x\prime (t). Then we can write an equation for y(t) in the form
y\prime (t) +
p\sum
j=1
aj(t)y(t - \tau j(t)) = \phi (t), t \in [0, \omega ],
y(tk) = \delta ky(tk - 0), k = 1, 2, . . . , r,
y(\zeta ) = 0, \zeta < 0.
(3.2)
Since y(t) satisfies a condition y(0) > 0, it is clear that there exists an interval [0, \mu ) such that
y(t) > 0 for t \in [0, \mu ), and, consequently, x(t) > 0 and x(t) increases at all the points t \not = tk for
t \in [0, \mu ).
In this case,
\sum p
j=1
bj(t)x(t - \theta j(t)) \leq 0, so it means that \phi (t) \geq 0 for t \in [0, \mu ). Then,
according to representation of solutions [7],
y(t) = y(0)C1(t, 0) +
t\int
0
C1(t, s)\phi (s)ds,
where C1(t, s) is the Cauchy function of (3.2), and we obtain y(t) > 0 for t \in [0, \omega ]. It is clear that
x\prime (t) > 0 and x(t) > 0 for t \in (0, \omega ].
Now we start to prove that [0, \omega ] is a semi-nonoscillation interval of (1.4) – (1.6). Let us assume
the contrary that [0, \omega ] is not a semi-nonoscillation interval. It means that one of the conditions is
fulfilled:
x\prime 1(\eta ) = 0, x1(\xi ) = 0, \eta < \xi ,
x1(\eta ) = 0, x\prime 1(\xi ) = 0, \eta < \xi .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
892 A. DOMOSHNITSKY, IU. MIZGIREVA, V. RAICHIK
η ξ
t
x
x1(t)
ξ
x1(t)
x
t
η
(a) (b)
Fig. 1. Cases x\prime
1 < 0 (a) and x\prime
1 = 0 (b).
We will prove that both of them are impossible.
1. Let us suppose that there exists a nontrivial solution x1(t) of the problem (1.4) – (1.6) such
that x1(\xi ) = 0, and x1(t) has zero of the derivative in the interval (0, \xi ). Without loss of generality
we can assume that x1(0) < 0, x\prime 1(\eta ) = 0, x\prime 1(t) < 0 for t \in [0, \eta ) and x\prime 1(t) > 0 for t \in (\eta , \omega ].
There are 2 possible situations for x1(t) that can be considered (see Fig. 1):
(a) x\prime 1(0) < 0 (see Fig. 1 (a)).
It is easy to show that in such situation the solution x1(t) will be negative and decreasing at all
the points t \not = tk, so it is impossible to achieve the boundary condition x1(\xi ) = 0.
Indeed, let us denote y(t) = x\prime 1(t). Then we can write an equation for y(t) in the form (3.2).
Since y(t) satisfies the condition y(0) < 0, it is clear that there exists an interval [0, \mu ) such that
y(t) < 0 for t \in [0, \mu ), and, consequently, x1(t) < 0 and x1(t) decreases at all the points t \not = tk for
t \in [0, \mu ).
In this case,
\sum p
j=1
bj(t)x1(t - \theta j(t)) \geq 0, so it means that \phi (t) \leq 0 for t \in [0, \mu ). Then,
according to representation of solutions [7],
y(t) = y(0)C1(t, 0) +
t\int
0
C1(t, s)\phi (s)ds,
where C1(t, s) is the Cauchy function of (3.2), and we obtain y(t) < 0 for t \in [0, \omega ]. Now it is clear
that x\prime 1(t) < 0 and x1(t) < 0 for t \in (0, \omega ].
We got a contradiction with the assumption that x1(\xi ) = 0.
(b) x\prime 1(0) = 0 (see Fig. 1 (b)).
It is easy to show that in such situation the solution x1(t) will be negative and decreasing at all
the points t \not = tk, so it will not be able to satisfy the boundary conditions x1(\xi ) = 0.
Since x1(0) < 0, it is clear that there exists an interval [0, \mu ) such that x1(t) < 0 for t \in [0, \mu ).
Let us denote y(t) = x\prime 1(t). Then we can write an equation for y(t) in the form (3.2).
In this case,
\sum p
j=1
bj(t)x1(t - \theta j(t)) \geq 0, so it means that \phi (t) \leq 0 for t \in [0, \mu ). Then,
according to the representation of solutions [7],
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
SEMI-NONOSCILLATION INTERVALS AND SIGN-CONSTANCY OF GREEN’S FUNCTIONS . . . 893
y(t) =
t\int
0
C1(t, s)\phi (s)ds,
where C1(t, s) is the Cauchy function of (3.2), and we obtain y(t) < 0 for t \in [0, \omega ]. Now it is clear
that x\prime 1(t) < 0 and x1(t) < 0 for t \in (0, \omega ].
We got a contradiction with the assumption that x1(\xi ) = 0.
2. Let us suppose that there exists a nontrivial solution x1(t) of the problem (1.4) – (1.6) such
that x1(\xi ) = \beta > 0, x\prime 1(\xi ) = 0, and x1(t) changes its sign in the interval (0, \xi ). Without the loss
of generality we can assume that x1(0) < 0, x1(\eta ) = 0, x1(t) < 0 for t \in [0, \eta ) and x1(t) > 0 for
t \in (\eta , \omega ].
There are 3 possible cases for x1(t) that can be considered (see Fig. 2):
(a) x\prime 1(0) < 0 (see Fig. 2 (a)).
This completely corresponds to the situation 1 (a). We have a contradiction with the boundary
condition x1(\xi ) = \beta > 0.
(b) x\prime 1(0) = 0 (see Fig. 2 (b)).
This completely corresponds to the situation 1 (b). We have a contradiction with the boundary
condition x1(\xi ) = \beta > 0.
(c) x\prime 1(0) > 0 (see Fig. 2 (c)).
The solution x1(t) is increasing, so it can satisfy the boundary conditions x1(\xi ) = \beta > 0,
x\prime 1(\xi ) = 0.
Denote a solution z(t) such that
z(t) =
x(\xi )
x1(\xi )
x1(t) - x(t).
It is clear that
z(0) =
x(\xi )
x1(\xi )
x1(0) - x(0) < 0,
z(\xi ) =
x(\xi )
x1(\xi )
x1(\xi ) - x(\xi ) = 0,
and
z\prime (\xi ) =
x(\xi )
x1(\xi )
x\prime 1(\xi ) - x\prime (\xi ) = - x\prime (\xi ) < 0.
In other words, the solution z(t) starts from a negative value at the point t = 0 and comes to
zero (from above) at the point t = \xi . Taking into account the fact that \gamma k > 0 and \delta k > 0, k = 1, r,
we see that the solutions cannot change sign at the points of impulses. This means that there exists
a point \mu \in (0, \xi ) such that z(\mu ) = 0. Thus, the solution z(t) has at least 2 zeros in the interval
(0, \omega ).
We have proven that the solution z(t) has at least 2 zeros in the interval [0, \omega ]. On the other
hand, we know that x(t), which is another linearly independent solution of (1.4) – (1.6), is positive
and increasing for t \in (0, \omega ]. Thus, x(t) has no zeros on (0, \omega ] in contradiction to Lemma 2.1
(see [10]).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
894 A. DOMOSHNITSKY, IU. MIZGIREVA, V. RAICHIK
t
(a)
t
(b)
t
(c)
Fig. 2. Examples of solution x1(t): cases x\prime
1 < 0 (a), x\prime
1 = 0 (b) and
x\prime
1 > 0 (c).
These contradictions show that every nontrivial solution of (1.4) – (1.6), having zero of derivative
on the interval (0, \omega ), does not have zero itself on this interval. So, according to the Definition 2.1,
the interval [0, \omega ] is a semi-nonoscillation interval.
Theorem 3.1 is proved.
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SEMI-NONOSCILLATION INTERVALS AND SIGN-CONSTANCY OF GREEN’S FUNCTIONS . . . 895
In Theorem 3.1 we have assumed that the Cauchy function C1(t, s) of the first order impulsive
equation (1.4) – (1.6) is positive. In the lemma below, we will formulate the results of [7] on the
conditions of positivity of Cauchy function of the first order impulsive differential equation
y\prime (t) + a1(t) y(t - \tau 1(t)) = 0, t \in [0, \omega ],
y(tk) = \delta ky(tk - 0), k = 1, 2, . . . , r,
y(\zeta ) = 0, \zeta < 0.
(3.3)
Lemma 3.1. Let \delta k < 1 for k = 1, . . . , r and the following condition be fulfilled:
1 + \mathrm{l}\mathrm{n}B(t)
e
\geq
t\int
m(t)
a+(s)ds, (3.4)
where B(t) =
\prod
k\in Dt
\delta k, Dt =
\bigl\{
i : ti \in [t - \tau 1(t), t]
\bigr\}
, a+(t) = \mathrm{m}\mathrm{a}\mathrm{x}\{ a1(t), 0\} and m(t) =
= \mathrm{m}\mathrm{a}\mathrm{x}\{ t - \tau 1(t), 0\} . Then Cauchy function of the first order impulsive differential equation (3.3) is
positive.
Using Lemma 3.1, we can reformulate Theorem 3.1 as follows.
Corollary 3.1. Assume that the following conditions are fulfilled:
(1) p = 1, a1(t) \geq 0, b1(t) \leq 0, t \in [0, \omega ];
(2) the Wronskian W (t) of the fundamental system of solutions of homogeneous equation (1.4) –
(1.6) satisfies the inequality W (t) \not = 0, t \in [0, \omega ];
(3) \delta k < 1 for k = 1, . . . , r and the condition (3.4) is fulfilled.
Then the interval [0, \omega ] is a semi-nonoscillation interval of (1.4) – (1.6).
In the case of p > 1 the following sufficient condition proven in [7] can be used.
Lemma 3.2. Let aj(t) \geq 0 for j = 1, . . . , p, \delta k < 1 for k = 1, . . . , r, and
\omega \int
0
p\sum
j=1
aj(s)ds <
r\prod
k=1
\delta k. (3.5)
Then Cauchy function of the first order impulsive differential equation (2.12) is positive.
Remark 3.2. In Lemma 3.2, the interval [0, \omega ] has to be small enough. In Lemma 3.1, the delay
has to be small enough.
Using Lemma 3.2, we can reformulate Theorem 3.1 as follows.
Corollary 3.2. Assume that the following conditions are fulfilled:
(1) aj(t) \geq 0, bj(t) \leq 0, j = 1, . . . , p, t \in [0, \omega ];
(2) the Wronskian W (t) of the fundamental system of solutions of homogeneous equation (1.4) –
(1.6) satisfies the inequality W (t) \not = 0, t \in [0, \omega ];
(3) \delta k < 1 for k = 1, . . . , r and the condition (3.5) is fulfilled.
Then the interval [0, \omega ] is a semi-nonoscillation interval of (1.4) – (1.6).
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896 A. DOMOSHNITSKY, IU. MIZGIREVA, V. RAICHIK
Consider the equation
y\prime (t) + a y(h(t)) = 0, t \in [0, \omega ],
y(tk) = \delta ky(tk - 0), k = 1, 2, . . . , r,
y(\zeta ) = 0, \zeta < 0,
(3.6)
where a is a positive constant and deviations are piecewise constant functions h(t) = tk, t \in
\in [tk, tk+1). Solving this equation on the intervals t \in [tk, tk+1), we can obtain the following fact.
Lemma 3.3. If a > 0, h(t) = tk, t \in [tk, tk+1), then the solution y(t) of (3.6) has the following
representation:
y(t) =
i\prod
j=1
\delta j
i\prod
j=1
\bigl(
1 - a(tj - tj - 1)
\bigr) \bigl(
1 - a(t - ti)
\bigr)
, t \in [ti, ti+1).
Remark 3.3. It is clear that Cauchy function C1(t, s) for the equation (3.6) satisfies the inequality
C1(t, s) > 0, if
a \mathrm{m}\mathrm{a}\mathrm{x}
j=1,...,r+1
(tj - tj - 1) < 1. (3.7)
Using Lemma 3.3 and Remark 3.3, we can reformulate Theorem 3.1 in the following way.
Corollary 3.3. Assume that the following conditions are fulfilled:
(1) p = 1, a1(t) = a \geq 0, b1(t) \leq 0, t - \tau 1(t) = tk, t \in [tk, tk+1), t \in [0, \omega ];
(2) the Wronskian W (t) of the fundamental system of solutions of homogeneous equation (1.4) –
(1.6) satisfies the inequality W (t) \not = 0, t \in [0, \omega ];
(3) \delta k < 1 for k = 1, . . . , r and the condition (3.7) is fulfilled.
Then the interval [0, \omega ] is a semi-nonoscillation interval of (1.4) – (1.6).
4. Semi-nonoscillation and sign-constancy of Green’s functions. In this section, we formulate
our main results.
Lemma 4.1. W (\xi ) \not = 0 if and only if the boundary-value problem (1.1) – (1.3) with boundary
conditions x(\xi ) = 0, x\prime (\xi ) = 0 is uniquely solvable.
Proof. The Wronskian W (\xi ) is a determinant of the system
c1\nu 1(\xi ) + c2C(\xi , 0) = 0,
c1\nu
\prime
1(\xi ) + c2C
\prime
t(\xi , 0) = 0.
(4.1)
Every solution of the homogeneous equation (1.4) – (1.6) has the form x(t) = c1\nu 1(t) + c2C(t, 0),
then a nontrivial solution of (1.1) – (1.3) with boundary conditions x(\xi ) = 0, x\prime (\xi ) = 0 exists if and
only if a nontrivial solution \{ c1, c2\} of the system (4.1) exists. This is equivalent to W (\xi ) = 0.
It is known from the general theory of functional differential equations [4] that the boundary-
value problem (1.1) – (1.3) with boundary conditions x(\xi ) = 0, x\prime (\xi ) = 0 is uniquely solvable (i.e.,
Green’s function G\xi (t, s) exists), if and only if the system (4.1) has only the trivial solution. This
means that W (\xi ) \not = 0.
Lemma 4.1 is proved.
Theorem 4.1. If aj(t) \geq 0, bj(t) \leq 0, then the following assertions are equivalent:
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SEMI-NONOSCILLATION INTERVALS AND SIGN-CONSTANCY OF GREEN’S FUNCTIONS . . . 897
s
t3
t2
t1
t1 t2 t3 t
–(t – s)
–(t – s)
–(t – s)
–(t – s)
ω
ω
0
Fig. 3. The Green’s function of impulsive equation (4.2) – (4.4) with boundary conditions x(\omega ) = 0, x\prime (\omega ) = 0.
(a) the Wronskian W (t) of the fundamental system of solutions of a homogeneous equation (1.4) –
(1.6) satisfies W (t) \not = 0, t \in [0, \omega ];
(b) Green’s function G\xi (t, s) is nonnegative for t, s \in [0, \xi ] for every 0 < \xi < \omega .
Proof. Firstly, let us prove the implication (b) \Rightarrow (a). Its proof is based on Lemma 4.1.
According to Lemma 4.1, it follows from the fact of the existence of Green’s function G\xi (t, s)
for every 0 < \xi < \omega , that the Wronskian W (t) \not = 0, t \in [0, \omega ].
Now let us prove the implication (a) \Rightarrow (b). If the Wronskian W (t) \not = 0, t \in [0, \omega ], then,
according to Lemma 4.1, Green’s function G\xi (t, s) exists for every 0 < \xi < \omega .
Consider the auxiliary boundary-value problem
x\prime \prime (t) = z(t), t \in [0, \omega ], (4.2)
x(tk) = \gamma kx(tk - 0), x\prime (tk) = \delta kx
\prime (tk - 0), k = 1, 2, . . . , r, (4.3)
x(\zeta ) = 0, x\prime (\zeta ) = 0, \zeta < 0, (4.4)
with the boundary conditions (2.11) for every 0 < \xi < \omega . Let us denote Green’s function for
boundary-value problem (4.2) – (4.4), (2.11) as G0\xi (t, s). It is nonnegative (see Fig. 3).
We can rewrite the problem (1.1) – (1.3), (2.11) as follows:
z(t) +
p\sum
j=1
aj(t)
\xi \int
0
G\prime
0\xi (t - \tau j(t), s)\chi [0,\xi ] (t - \tau j(t)) z(s)ds+
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898 A. DOMOSHNITSKY, IU. MIZGIREVA, V. RAICHIK
+
p\sum
j=1
bj(t)
\xi \int
0
G0\xi (t - \theta j(t), s)\chi [0,\xi ] (t - \theta j(t)) z(s)ds = f(t), (4.5)
where the characteristic function \chi [0,\xi ](t) is defined by (2.13).
Denote
(K0\xi z)(s) =
\xi \int
0
\Biggl[
-
p\sum
j=1
aj(t)G
\prime
0\xi (t - \tau j(t), s)\chi [0,\xi ] (t - \tau j(t)) -
-
p\sum
j=1
bj(t)G0\xi (t - \theta j(t), s)\chi [0,\xi ] (t - \theta j(t))
\Biggr]
z(s)ds.
For aj(t) \geq 0, bj(t) \leq 0, the operator K0\xi is positive. According to Lemma 4.1, if the
Wronskian W (t) \not = 0, t \in [0, \omega ], then the problem (1.1) – (1.3) with condition (2.11) is uniquely
solvable for every f \in L\infty and Green’s function G\xi (t, s) exists for every 0 < \xi < \omega . The fact of
the unique solvability for every 0 < \xi < \omega implies that the spectral radius \rho (K0\xi ) < 1 (see [18]).
So, the solution of (4.5) can be represented in the form
z = (I - K0\xi )
- 1f =
\left[ \infty \sum
j=0
Kj
0\xi
\right] f,
where I : L\infty [0, \xi ] \rightarrow L\infty [0, \xi ] is a unit operator acting in the space of essentially bounded functions
f : [0, \xi ] \rightarrow \BbbR .
The solution x(t) of the boundary-value problem (1.1) – (1.3), (2.11) can be written in the form
x =
\left( G0\xi
\infty \sum
j=0
Kj
0\xi
\right) f,
where G0\xi
\sum \infty
j=0
Kj
0\xi is Green’s operator for (1.1) – (1.3) with boundary conditions (2.11). From
nonnegativity of Green’s function G0\xi (t, s) and positivity of operator K0\xi , it follows that Green’s
function G\xi (t, s) for (1.1) – (1.3), (2.11) is nonnegative for (t, s) \in [0, \xi ]\times [0, \xi ] for every 0 < \xi < \omega .
Theorem 4.1 is proved.
We use the semi-nonoscillation intervals in the proof of the following assertion on nonpositivity
of Green’s functions (2.7) – (2.10).
Theorem 4.2. Assume that the following conditions are fulfilled:
(1) aj(t) \geq 0, bj(t) \leq 0, j = 1, . . . , p, t \in [0, \omega ];
(2) the Wronskian W (t) of the fundamental system of solutions of the homogeneous equation
(Lx)(t) = 0, (1.2) – (1.3) satisfies the inequality W (t) \not = 0, t \in [0, \omega ];
(3) the Cauchy function C1(t, s) of the first order equation (2.12) is positive for 0 \leq s \leq t \leq \omega .
Then Green’s functions Gi(t, s), i = 1, 3, are nonpositive for t, s \in [0, \omega ] and under the additional
condition
\sum p
j=1
bj(t)\chi [0,\omega ](t - \theta j(t)) \not \equiv 0, t \in [0, \omega ], Green’s function G4(t, s) is nonpositive for
t, s \in [0, \omega ].
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SEMI-NONOSCILLATION INTERVALS AND SIGN-CONSTANCY OF GREEN’S FUNCTIONS . . . 899
x1(t)
x2(t) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t
0.0
0.2
0.4
0.6
0.8
x(t)
(a) (b)
Fig. 4. Calculating the Wronskian.
Proof. All the conditions of Theorem 3.1 are fulfilled. According to Theorem 3.1, the interval
[0, \omega ] is a semi-nonoscillation one. It was proven in Theorem 4.1 that the inequality W (t) \not = 0 for
t \in [0, \omega ], i.e., the condition 2 of Theorem 4.2, is equivalent to the condition 3 of Lemma 2.1. Now
we see that all the conditions of Lemma 2.1 are fulfilled. According to Lemma 2.1, Green’s functions
Gi(t, s), i = 1, 4, are nonpositive.
Theorem 4.1 is proved.
Example 4.1. Let us consider the differential equation
x\prime \prime (t)+x\prime (h(t)) - x(h(t)) = f(t), t \in [0, 1.6],
t1 = 0.4, \gamma 1 = 0.8, \delta 1 = 0.7,
t2 = 1, \gamma 2 = 0.9, \delta 2 = 0.95.
(4.6)
Let us assume that the delays have the following forms:
h(t) = tk, t \in [tk, tk+1), k = 0, 1, 2.
According to Lemma 3.3 and Remark 3.3, the Cauchy function of the first order impulsive
equation
y\prime (t) + y(h(t)) = 0, t \in [0, 1.6],
t1 = 0.4, \gamma 1 = 0.8, \delta 1 = 0.7,
t2 = 1, \gamma 2 = 0.9, \delta 2 = 0.95,
satisfies the inequality C1(t, s) \geq 0, if
\mathrm{m}\mathrm{a}\mathrm{x}
k=1,...,3
(tk - tk - 1) < 1,
where t0 = 0, t3 = 1.6.
Thus, in our example, the condition (3.7) holds. Let us solve impulsive equation (4.6) with the
conditions x1(0) = 1, x\prime 1(0) = 0 and x2(0) = 0, x\prime 2(0) = 1. We obtain
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
900 A. DOMOSHNITSKY, IU. MIZGIREVA, V. RAICHIK
x1(t) =
\left\{
0, t < 0,
0.5t2 + 1, t \in [0, 0.4),
0.292(t - 0.4)2 + 0.28t+ 0.752, t \in [0.4, 1),
0.212(t - 1.0)2 + 0.599t+ 0.425, t \in [1, 1.6),
x2(t) =
\left\{
0, t < 0,
- 0.5t2 + t, t \in [0, 0.4),
- 0.082(t - 0.4)2 + 0.42t+ 0.088, t \in [0.4, 1),
0.063(t - 1.0)2 + 0.306t+ 0.125, t \in [1, 1.6),
see Fig. 4 (a).
Calculating the Wronskian, we obtain
W (t) =
\left\{
0, t < 0,
- 0.5t2 - t+ 1, t \in [0, 0.4),
- 0.146t2 - 0.175t+ 0.384, t \in [0.4, 1),
- 0.027t2 + 0.082, t \in [1, 1.6),
(4.7)
see Fig. 4 (b).
Thus, all the conditions of the Theorem 4.2 are fulfilled. So, according to Theorem 4.2, Green’s
functions Gi(t, s), i = 1, 4, are nonpositive.
References
1. R. P. Agarwal, L. Berezansky, E. Braverman, A. Domoshnitsky, Nonoscillation theory of functional differential
equations with applications, Springer (2012).
2. R. P. Agarwal, D. O’Regan, Multiple nonnegative solutions for second order impulsive differential equations, Appl.
Math. and Comput., 114, № 1, 51 – 59 (2000).
3. N. V. Azbelev, About zeros of solutions of linear differential equations of the second order with delayed argument,
Differents. Uravn., 7, № 7, 1147 – 1157 (1971).
4. N. V. Azbelev, V. P. Maksimov, L. F. Rakhmatullina, Introduction to the theory of functional differential equations:
methods and applications, Contemp. Math. and Appl., vol. 3, Hindawi, New York (2007).
5. A. A. Boichuk, A. M. Samoilenko, Generalized inverse operators and Fredholm boundary-value problems, VSP,
Utrecht; Boston (2004).
6. A. Domoshnitsky, Wronskian of fundamental system of delay differential equations, Funct. Different. Equat., 7,
445 – 467 (2002).
7. A. Domoshnitsky, M. Drakhlin, Nonoscillation of first order impulsive differential equations with delay, J. Math.
Anal. and Appl., 206, № 1, 254 – 269 (1997).
8. A. Domoshnitsky, G. Landsman, Semi-nonoscillation intervals in the analysis of sign constancy of Green’s functions
of Dirichlet, Neumann and focal impulsive problems, Adv. Differerence Equat., 81, № 1 (2017); DOI: 10.1186/s13662-
017-1134-1.
9. A. Domoshnitsky, G. Landsman, S. Yanetz, About sign-constancy of Green’s functions of one-point problem for
impulsive second order delay equations, Funct. Different. Equat., 21, № 1-2, 3 – 15 (2014).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
SEMI-NONOSCILLATION INTERVALS AND SIGN-CONSTANCY OF GREEN’S FUNCTIONS . . . 901
10. A. Domoshnitsky, V. Raichik, The Sturm separation theorem for impulsive delay differential equations, Tatra Mt.
Math. Publ., 71, № 1, 65 – 70 (2018).
11. A. Domoshnitsky, I. Volinsky, About differential inequalities for nonlocal boundary value problems with impulsive
delay equations, Math. Bohem., 140, № 2, 121 – 128 (2015).
12. A. Domoshnitsky, I. Volinsky, About positivity of Green’s functions for nonlocal boundary value problems with impulsi-
ve delay equations, Sci. World J. (TSWJ): Math. Anal., Article ID 978519 (2014); DOI: doi.org/10.1155/2014/978519.
13. A. Domoshnitsky, I. Volinsky, R. Shklyar, About Green’s functions for impulsive differential equations, Funct.
Different. Equat., 20, № 1-2, 55 – 81 (2013).
14. M. Feng, D. Xie, Multiple positive solutions of multi-point boundary value problem for second-order impulsive
differential equations, J. Math. Anal. and Appl., 223, № 1, 438 – 448 (2009).
15. L. Hu, l. Liu, Y. Wu, Positive solutions of nonlinear singular two-point boundary value problems for second-order
impulsive differential equations, Appl. Math. and Comput., 196, № 2, 550 – 562 (2008).
16. T. Jankowsky, Positive solutions to second order four-point boundary value problems for impulsive differential
equations, Appl. Math. and Comput., 202, № 2, 550 – 561 (2008).
17. D. Jiang, X. Lin, Multiple positive solutions of Dirichlet boundary value problems for second order impulsive
differential equations, J. Math. Anal. and Appl., 321, № 2, 501 – 514 (2006).
18. M. A. Krasnosel’skii, G. M. Vainikko, P. P. Zabreyko, Y. B. Ruticki, V. Y. Stetsenko, Approximate solution of operator
equations, Nauka, Moscow (1969) (in Russian).
19. S. M. Labovskii, Condition of nonvanishing of Wronskian of fundamental system of linear equation with delayed
argument, Differents. Uravn., 10, № 3, 426 – 430 (1974).
20. V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, World Sci., Singapore
(1989).
21. E. K. Lee, Y. H. Lee, Multiple positive solutions of singular two point boundary value problems for second order
impulsive differential equations, Appl. Math. and Comput., 158, № 3, 745 – 759 (2004).
22. Iu. Mizgireva, On positivity of Green’s functions of two-point impulsive problems, Funct. Different. Equat., № 3-4
(2018).
23. S. G. Pandit, S. G. Deo, Differential systems involving impulses, Lect. Notes Math., vol. 954, Springer-Verlag, New
York (1982).
24. A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, World Sci., Singapore (1992).
25. S. G. Zavalishchin, A. N. Sesekin, Dynamic impulse systems: theory and applications, Math. and Appl., 394, Kluwer,
Dordrecht (1997).
Received 12.11.18,
after revision — 15.01.19
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| id | umjimathkievua-article-473 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:55Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/01/b3f0a6c68638561facc3124751d52001.pdf |
| spelling | umjimathkievua-article-4732025-03-31T08:47:53Z Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems Domoshnitsky, A. Mizgireva, Iu. Raichik, V. Домошнитский, Александр Mizgireva, Iuliia Domoshnitsky, A. Mizgireva, Iu. Raichik, V. second order impulsive differential equations semi-nonoscillation intervals second order impulsive differential equations semi-nonoscillation intervals UDC 517.9 We consider the second order impulsive differential equation with delays&nbsp; $$(Lx)(t)\equiv x''(t) + \sum\limits _{j = 1}^{p} a_j(t) x'(t-\tau_j(t)) + \sum\limits _{j = 1}^{p} b_j(t) x(t-\theta_j(t)) = f(t),&nbsp; \quad t \in [0, \omega],&nbsp;$$&nbsp;$$x(t_k) = \gamma_k x(t_k-0), \quad x'(t_k) = \delta_k x'(t_k-0),\quad&nbsp; k = 1, 2, \ldots , r,$$ where $\gamma_k &gt; 0,$ $\delta_k &gt;0$ for $k = 1, 2, \ldots , r.$&nbsp;In this paper, we obtain the conditions of semi-nonoscillation for the corresponding homogeneous equation on the interval $[0, \omega].$&nbsp;&nbsp;Using these results, we formulate theorems on sign-constancy of Green's functions for two-point impulsive boundary-value problems in terms of differential inequalities.&nbsp; UDC 517.9 Інтервали майже вiдсутностi осциляцiй та збереження знаку функцiй Грiна двоточкових iмпульсних крайових задач Розглядається імпульсне диференціальне рівняння другого порядку із запізненням $$(Lx)(t)\equiv x''(t) + \sum\limits _{j = 1}^{p} a_j(t) x'(t-\tau_j(t)) + \sum\limits _{j = 1}^{p} b_j(t) x(t-\theta_j(t)) = f(t),&nbsp; \quad t \in [0, \omega], $$&nbsp;$$x(t_k) = \gamma_k x(t_k-0), \quad x'(t_k) = \delta_k x'(t_k-0),\quad&nbsp; k = 1, 2, \ldots , r,$$ де $\gamma_k &gt; 0,$ $\delta_k &gt;0$ для $k = 1, 2, \ldots , r.$&nbsp;Знайдено умови майже відсутності коливань для відповідного однорідного рівняння на інтервалі $[0, \omega].$&nbsp;&nbsp;За допомогою цих результатів сформульовано теореми про збереження знака функцій Гріна двоточкових імпульсних крайових задач у термінах диференціальних нерівностей.&nbsp; Institute of Mathematics, NAS of Ukraine 2021-07-20 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/473 10.37863/umzh.v73i7.473 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 7 (2021); 887 - 901 Український математичний журнал; Том 73 № 7 (2021); 887 - 901 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/473/9085 Copyright (c) 2021 A. Domoshnitsky, Iu. Mizgireva, V. Raichik |
| spellingShingle | Domoshnitsky, A. Mizgireva, Iu. Raichik, V. Домошнитский, Александр Mizgireva, Iuliia Domoshnitsky, A. Mizgireva, Iu. Raichik, V. Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems |
| title | Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems |
| title_alt | Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems |
| title_full | Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems |
| title_fullStr | Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems |
| title_full_unstemmed | Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems |
| title_short | Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems |
| title_sort | semi-nonoscillation intervals and sign-constancy of green’s functions of two-point impulsive boundary-value problems |
| topic_facet | second order impulsive differential equations semi-nonoscillation intervals second order impulsive differential equations semi-nonoscillation intervals |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/473 |
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