Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations
In 1997, A. H. Nasr provided necessary and sufficient conditions for the oscillation of the equation $$x''(t) + p(t) |x(g(t))|^{\eta} \text{sgn} (x(g(t))) = e(t),$$ where $\eta > 0$, $p$, and $g$ are continuous functions on $[0, \infty)$ such that $p(t) \geq 0,\;\; g(t) \leq...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508595174506496 |
|---|---|
| author | Cheng, Sui Sun Shao, Yuan Huang Ченг, Суй Сун Хуан, Шао Юань |
| author_facet | Cheng, Sui Sun Shao, Yuan Huang Ченг, Суй Сун Хуан, Шао Юань |
| author_sort | Cheng, Sui Sun |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:32:05Z |
| description | In 1997, A. H. Nasr provided necessary and sufficient conditions for the oscillation of the equation
$$x''(t) + p(t) |x(g(t))|^{\eta} \text{sgn} (x(g(t))) = e(t),$$
where $\eta > 0$, $p$, and $g$ are continuous functions on $[0, \infty)$ such that $p(t) \geq 0,\;\; g(t) \leq t,\;\; g'(t) \geq \alpha > 0$, and $\lim_{t \rightarrow \infty} g(t) = \infty$
It is important to note that the condition $g'(t) \geq \alpha > 0$ is required.
In this paper, we remove this restriction under the superlinear assumption $\eta > 0$.
Infact, we can do even better by considering impulsive differential equations with delay and obtain necessary and sufficient conditions
for the existence of nonoscillatory solutions and also a comparison theorem that enables us to apply
known oscillation results for impulsive equations without forcing terms to yield oscillation criteria for our equations. |
| first_indexed | 2026-03-24T02:27:42Z |
| format | Article |
| fulltext |
UDC 517.9
Shao Yuan Huang, Sui Sun Cheng (Tsing Hua Univ., Taiwan)
COMPARISON THEOREMS AND NECESSARY/SUFFICIENT CONDITIONS
FOR EXISTENCE OF NONOSCILLATORY SOLUTIONS
OF FORCED IMPULSIVE DELAY DIFFERENTIAL EQUATIONS
ТЕОРЕМИ ПОРIВНЯННЯ ТА НЕОБХIДНI/ДОСТАТНI УМОВИ IСНУВАННЯ
НЕОСЦИЛЯЦIЙНИХ РОЗВ’ЯЗКIВ ЗБУРЕНИХ IМПУЛЬСНИХ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ IЗ ЗАПIЗНЕННЯМ
In 1997, A. H. Nasr provided necessary and sufficient conditions for the oscillation of the equation
x′′(t) + p(t) |x(g(t))|η sgn (x(g(t))) = e(t),
where η > 0, p, and g are continuous functions on [0,∞) such that p(t) ≥ 0, g(t) ≤ t, g′(t) ≥ α > 0, and limt→∞ g(t) =
= ∞. It is important to note that the condition g′(t) ≥ α > 0 is required. In this paper, we remove this restriction under
the superlinear assumption η > 1. Infact, we can do even better by considering impulsive differential equations with delay
and obtain necessary and sufficient conditions for the existence of nonoscillatory solutions and also a comparison theorem
that enables us to apply known oscillation results for impulsive equations without forcing terms to yield oscillation criteria
for our equations.
У 1997 роцi, А. Х. Наср отримав необхiднi та достатнi осциляцiйнi умови для рiвняння
x′′(t) + p(t) |x(g(t))|η sgn (x(g(t))) = e(t),
де η > 0, p та g — неперервнi функцiї на [0,∞) такi, що p(t) ≥ 0, g(t) ≤ t, g′(t) ≥ α > 0 та limt→∞ g(t) =∞. Слiд
зауважити, що необхiдною тут є умова g′(t) ≥ α > 0. У данiй статтi ми усуваємо це обмеження при суперлiнiйному
припущеннi η > 1. Насправдi, можна отримати навiть кращий результат, розглядаючи iмпульснi диференцiальнi
рiвняння з запiзненням, i встановити необхiднi та достатнi умови iснування неосциляцiйних розв’язкiв, а також
теорему порiвняння, яка дає змогу застосувати вiдомi осциляцiйнi результати для iмпульсних рiвнянь без збурюючих
членiв, щоб отримати осциляцiйнi критерiї для наших рiвнянь.
1. Introduction. In 1997, A. H. Nasr in [1] provided necessary and sufficient conditions for the
oscillation of the equation
x′′(t) + p(t) |x(g(t))|η sgn (x(g(t))) = e(t), (1)
where η > 0, p and g are continuous functions on [0,∞) such that p(t) ≥ 0, g(t) ≤ t, g′(t) ≥ α > 0
and limt→∞ g(t) = ∞. Under a nice assumption on the function e (that the solution z of (5) is
oscillatory and (22) holds), it is stated in reference [1] that the following conclusions hold: for η > 1,
equation (1) is oscillatory if, and only if,
∫ ∞
0
tp(t)dt = ∞; and for 0 < η < 1, equation (1) is
oscillatory if, and only if
∫ ∞
0
trp(t)dt =∞. These conclusions extend those in [2] in which the well
known Emden – Fowler equation without delay is studied.
It is important to note that the condition “g′(t) ≥ α > 0” is needed in [1]. However, in [3], the
author removes the restriction for the sublinear case 0 < η < 1. In this paper, we intend to improve
the same restriction for the superlinear case η > 1 (see Corollary 3, or Theorems 1 and 4 below).
Indeed, we can do even better by considering impulsive differential equations with delay. More
specifically, we obtain necessary and sufficient conditions for the existence of nonoscillatory solutions
c© SHAO YUAN HUANG, SUI SUN CHENG, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 9 1233
1234 SHAO YUAN HUANG, SUI SUN CHENG
and also a comparison theorem which enables us to apply known oscillation results (see, e.g., [4]) for
impulsive equations without forcing terms to yield oscillation criteria for our equations (an example
is illustrated in the last section).
To this end, we first recall some usual notations. R and N denote the set of real numbers and
positive integers respectively. R+ denotes the interval (0,+∞). Assume I1 and I2 are any two
intervals in R, we define
ALC(I1, I2) =
{
ϕ : I1 → I2 : ϕ is continuous almost everywhere (a.e.) in I1
with discontinuities of first kind
}
,
PC(I1, I2) =
{
ϕ ∈ ALC(I1, I2) : ϕ is continuous in each interval I1 ∩ (tk, tk+1], k ∈ N0
}
,
and
PC ′(I1, I2) =
{
ϕ ∈ PC(I1, I2) : ϕ is continuously differentiable a.e. in I1
}
.
We let
Υ = {t1, t2, . . .}
be a set of real numbers such that 0 = t0 < t1 < t2 < . . . and limk→∞ tk = +∞. Also, x′(t) will
be used to denote the left derivative of the function x(t) at t. We investigate the following nonlinear
delay differential systems ‘with impulsive effects’(
r(t)x′(t)
)′
+ F (t, x(g(t))) = e(t), t ∈ [0,∞)\Υ, (2)
x(t+k ) = akx(tk), k ∈ N, (3)
x′(t+k ) = bkx
′(tk), k ∈ N, (4)
under some of the following conditions:
(A1) For t ≥ 0, the function F (t, µ) is continuous on R with µF (t, µ) ≥ 0 for µ 6= 0, and
for µ ∈ R, the function F (t, µ) belongs to ALC([0,∞),R). Furthermore, F (t, µ2) ≥ F (t, µ1) for
t ≥ 0 and µ2 ≥ µ1;
(A2) g is a continuous function on [0,∞) with g(t) ≤ t for t ≥ 0 and limt→∞ g(t) = +∞;
(A3) 0 < t1 < t2 < . . . are fixed numbers with limk→∞ tk = +∞;
(A4) for each k ∈ N, ak > 0 and bk > 0;
(A5) r is a positive and differentiable function on [0,∞);
(A6) e is a function on [0,∞) continuous a.e.;
(A7) there are M > 0 and m > 0 such that m ≤ A(s, t) ≤M for t ≥ s ≥ 0 where
A(s, t) =
∏
s≤tk<t
ak if [s, t) ∩Υ 6= ∅,
1 if [s, t) ∩Υ = ∅.
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COMPARISON THEOREMS AND NECESSARY/SUFFICIENT CONDITIONS FOR EXISTENCE . . . 1235
Let σ ≥ 0 be given. We define rσ = mint≥σ g(t) and
B(s, t) =
∏
s≤tk<t
bk if [s, t) ∩Υ 6= ∅,
1 if [s, t) ∩Υ = ∅
for t ≥ s ≥ 0.
Definition 1. Let σ ≥ 0. For any φ ∈ PC ′([rσ, σ],R), a function x ∈ PC ′([rσ,∞),R) is
said to be a solution of system (2) – (4) on [σ,∞) satisfying the initial value condition
x(t) = φ(t), t ∈ [rσ, σ],
if the following conditions are satisfied:
(i) x′ ∈ PC ′([σ,∞),R);
(ii) x satisfies (2) for a.e. t ≥ σ;
(iii) x satisfies (3) and (4) for t ≥ σ.
Definition 2. Let x = x(t) be a real function defined for all sufficiently large t. We say that
x is eventually positive (or negative) if there exists a number T such that x(t) > 0 (respectively
x(t) < 0) for every t ≥ T. We say that x is nonoscillatory if x(t) is eventually positive or eventually
negative. Otherwise, x is said to be oscillatory.
In the subsequent discussions, we assume that there exists a solution z of the system
(r(t)z′(t))′ = e(t), t ∈ [0,∞)\Υ,
z(t+k ) = akz(tk), k ∈ N,
z′(t+k ) = bkz
′(tk), k ∈ N,
(5)
on [τ,∞) for some τ ≥ 0. Let T ≥ 0 and ϕ ∈ PC([rT , T ],R). For δ ∈ PC([T,∞),R), we define
a function wϕ(δ) by
wϕ(δ)(t) =
δ(g(t)) if g(t) > T,
ϕ(g(t)) if rT ≤ g(t) ≤ T
for t ≥ T.
This paper is mainly concerned with oscillation of impulsive differential equations, but for more
general background material, the reader is referred to [5 – 8].
2. Main results. We begin with a simple comparison principle.
Lemma 1. Assume that (A1) – (A6) hold, that the solution z of (5) is oscillatory, and
∞∫
t0
B(t0, s)
A(t0, s)r(s)
ds =∞ (6)
for any t0 ≥ 0. Let x be an eventually positive solution of system (2) – (4). Then x(t) > z(t) and
x′(t) ≥ z′(t) eventually.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 9
1236 SHAO YUAN HUANG, SUI SUN CHENG
Proof. Without loss of generality, we may assume that τ = 0, and x(t) > 0 for t ≥ r0. Let
y(t) = x(t)− z(t) for t ≥ 0. So the function y satisfies(
r(t)y′(t)
)′
+ F (t, x(g(t))) = 0, a.e. t ≥ 0, (7)
and
y(t+k ) = aky(tk) and y′(t+k ) = bky
′(tk), k ∈ N. (8)
By (7), we see that (r(t)y′(t))′ ≤ 0 for a.e. t ≥ 0. Assume that there exists T > 0 such that
(r(T )y′(T ))′ exists and y′(T ) < 0. By (8),
r(t)y′(t) ≤ B(T, t)r(T )y′(T ) < 0, a.e. t ≥ T. (9)
Dividing (9) by r(t)A(T, t), and then integrating the subsequent inequalities from T to t, we obtain
y(t) ≤ A(T, t)
y(T ) + r(T )′y′(T )
t∫
T
B(T, s)
A(T, s)r(s)
ds
, t ≥ T.
In view of (6), y(t) < 0 eventually. This is a contradiction since x(t) > 0 eventually. So y′(t) ≥ 0
eventually. We note that it is impossible that y(t) ≤ 0 eventually because of x(t) > 0 eventually. So
there exists sufficiently large T2 such that y(T2) > 0, then
y(t) ≥ A(T2, t)y(T2) > 0, t ≥ T2,
which implies that y(t) > 0 eventually.
Lemma 1 is proved.
Remark 1. If e(t) = 0 eventually, we may assume without loss of generality that the function
z is the trivial function. By Lemma 1, we may see that the derivative of any eventually positive
solution of system (2) – (4) is eventually nonnegative.
Theorem 1. Assume that (A1) – (A7) hold, that the solution z of (5) is bounded, and that
∞∫
ε
1
r(s)
∞∫
s
F (v, c)
B(s, v)
dvds <∞ (10)
for any c > 0 and some ε ≥ 0. Then system (2) – (4) has an eventually positive solution x which is
bounded.
Proof. Without loss of generality, we may assume that |z(t)| ≤M for t > τ. Let
d = M
(
M + 1
m
+ 2
)
.
Clearly, d > 1. By (10), there exists T ∈ Υ such that T > max {τ, ε} and
∞∫
T
1
r(s)
∞∫
s
F (v, d)
B(s, v)
dvds ≤ 1. (11)
Let
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COMPARISON THEOREMS AND NECESSARY/SUFFICIENT CONDITIONS FOR EXISTENCE . . . 1237
X1 =
{
δ ∈ PC ([T,∞),R) : 1 ≤ δ(t) ≤ d for t ≥ T
}
. (12)
Let ϕ(t) = 1 for rT ≤ t ≤ T. We define the operator H1 on X1 by
H1(δ)(t) = A(T, t)
M + 1
m
+ z(t) +
t∫
T
A(s, t)
r(s)
∞∫
s
F (v, wϕ(δ)(v))
B(s, v)
dvds (13)
for t ≥ T. Obviously, H1(δ) ∈ PC([T,∞),R). By the definition of wϕ(δ), we may see that
1 ≤ w(δ)(t) ≤ d for t ≥ T. By (11),
H1(δ)(t) ≥ m
M + 1
m
−M = 1
and
H1(δ)(t) ≤M
M + 1
m
+M +M
t∫
T
1
r(s)
∞∫
s
F (v, d)
B(s, v)
dvds ≤M
(
M + 1
m
+ 2
)
= d (14)
for t ≥ T. So H1(X1) ⊆ X1. We shall use the Knaster – Tarski fixed point theorem to prove that
H1 has a fixed point in X1. We first define a relation in X1. If δ1 and δ2 belong to X1, let us
say that δ1 ≤ δ2 if and only if δ1(t) ≤ δ2(t) a.e. on [T,∞). Clearly, X1 is a complete lattice.
Given δ1, δ2 ∈ X1 with δ1 ≤ δ2. Then wϕ(δ1)(t) ≤ wϕ(δ2)(t) for a.e. t ≥ T, which follows that
F (t, wϕ(δ1)(t)) ≤ F (t, wϕ(δ2)(t)) for a.e. t ≥ T. Then H1(δ1)(t) ≤ H1(δ2)(t) for a.e. t ≥ T. So
H1 is increasing in X1. By the Knaster – Tarski fixed point theorem, there exists θ1 ∈ X1 such that
H1(θ1) = θ1. Let
x(t) =
θ1(t) if t > T,
1 if T ≥ t ≥ rT .
Clearly, x ∈ PC ′([T,∞), [1, d]). Let T1 > 0 such that rT1 > T. We note that x(g(t)) = wϕ(θ1)(t)
for t ≥ T1. In view of H1(θ1) = θ1, we have
x′(t) = z′(t) +
B(T1, t)
r(t)
∞∫
t
F (s, x(g(s)))
B(T1, s)
ds, t ≥ T1, (15)
which leads us to x′ ∈ PC ′([T1,∞),R) and(
r(t)x′(t)
)′
+ F (t, x(g(t))
)
= e(t), a.e. t > T1.
For any tk ≥ T1, by H1(θ1) = θ1 and (15),
x(t+k ) = akx(tk) and x′(t+k ) = bkx
′(tk).
Then x is a bounded and positive solution of system (2) – (4) on [T1,∞).
Theorem 1 is proved.
As a direct consequence, we have the following dual conclusion.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 9
1238 SHAO YUAN HUANG, SUI SUN CHENG
Corollary 1. Assume that (A1) – (A7) hold, that the solution z of (5) is bounded, and that
∞∫
ε
1
r(s)
∞∫
s
F (v, τ)
B(s, v)
dvds > −∞ (16)
for any τ < 0 and some ε ≥ 0. Then system (2) – (4) has an eventually negative solution x which is
bounded.
Lemma 2. Assume that (A1) – (A6) hold. If the system(
r(t)u′(t)
)′
+ F (t, u(g(t))) ≤ 0, t ∈ [0,∞)\Υ,
u(t+k ) = aku(tk), k ∈ N, (17)
u′(t+k ) = bku
′(tk), k ∈ N,
has an eventually positive solution u with u(t)u′(t) ≥ 0 eventually, then(
r(t)u′(t)
)′
+ F (t, u(g(t))) = 0, t ∈ [0,∞)\Υ, (18)
u(t+k ) = aku(tk), k ∈ N, (19)
u′(t+k ) = bku
′(tk), k ∈ N, (20)
has an eventually positive solution solution ũ with ũ(t) ũ′(t) ≥ 0 eventually.
Proof. There is T > 0 such that u(t) > 0 and u′(t) ≥ 0 for t ≥ rT . For any d ≥ t ≥ T, we
divide (17) by B(T, t), and then integrate from t to d. We have
r(d)u′(d)
B(T, d)
− r(t)u′(t)
B(T, t)
+
d∫
t
F (s, u(g(s)))
B(T, s)
ds ≤ 0.
Since u′(d) ≥ 0 and d is arbitrary, we may see that
u′(t) ≥ 1
r(t)
∞∫
t
F (s, u(g(s)))
B(t, s)
ds
for t ≥ T. Again, we divide the above inequality by A(T, t) and then integrate from T to t. Then we
have
u(t) ≥ A(T, t)
u(T ) +
t∫
T
1
A(T, s)r(s)
∞∫
s
F (v, u(g(v)))
B(s, v)
dvds
(21)
for t ≥ T. Let
X2 =
{
δ ∈ PC([T,∞),R) : A(T, t)u(T ) ≤ δ(t) ≤ u(t) for t ≥ T
}
and we define an operator H2 on X2 by
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COMPARISON THEOREMS AND NECESSARY/SUFFICIENT CONDITIONS FOR EXISTENCE . . . 1239
H2(δ)(t) = A(T, t)
u(T ) +
t∫
T
1
A(T, s)r(s)
∞∫
s
F (v, wu(δ)(v)))
B(s, v)
dvds
for t ≥ T. Clearly, X2 is nonempty because u ∈ X2. We impose in X2 the same order relation
imposed in the set X1. Then X2 is a complete lattice. Given δ1, δ2 ∈ X2 with δ1 ≤ δ2. We note that
0 < wu(δ1)(t) ≤ wu(δ2)(t) ≤ u(g(t))
for t ≥ T. By assumption,
F (t, wu(δ1)(t)) ≤ F (t, wu(δ2)(t)) ≤ F (t, u(g(t))),
where t ≥ T. By (21),
A(T, t)u(T ) ≤ H2(δ1)(t) ≤ H2(δ2)(t) ≤ u(t)
for t ≥ T. So H2(X2) ⊆ X2 and H2 is increasing on X2. By the Knaster – Tarski fixed point
Theorem, there exists θ2 ∈ X2 such that H2(θ2) = θ2. Let
µ̃(t) =
θ2(t) if t > T,
u(t) if T ≥ t ≥ rT .
Similar to the proof of Theorem 1, we may check that µ̃ is an eventually positive solution of
system (18) – (20) with µ̃(t)µ̃′(t) ≥ 0 eventually.
Lemma 2 is proved.
We now compare forced impulsive equations and unforced impulsive equations.
Theorem 2. Assume that (A1) – (A6) and (6) hold and the solution z of (5) is oscillatory.
Assume that there exist two sequences {sn}n∈N and {s̃n}n∈N such that
z(sn) = inf
{
z(t)
A(sn, t)
: t ≥ sn
}
and z(s̃n) = sup
{
z(t)
A(s̃n, t)
: t ≥ s̃n
}
. (22)
If the system (2) – (4) has a nonoscillatory solution x, then the system(
r(t)u′(t)
)′
+ F (t, u(g(t))) = 0, t ∈ [0,∞)\Υ, (23)
u(t+k ) = aku(tk), k ∈ N, (24)
u′(t+k ) = bku
′(tk), k ∈ N, (25)
has a nonoscillatory solution u such that u(t)u′(t) ≥ 0 eventually. Furthermore, u(t) is bounded if
x(t) is bounded and (A7) holds.
Proof. We first assume that x is an eventually positive solution of (2) – (4). By Lemma 1, x(t) >
> z(t) and x′(t) ≥ z′(t) eventually. Without loss of generality, we may assume that τ = 0, x(t) > 0,
x(t) > z(t) and x′(t) ≥ z′(t) for t ≥ s1. Let y(t) = x(t) − z(t) and v(t) = y(t) + A(s1, t)z(s1)
for t ≥ s1. Clearly, v, v′ ∈ PC ′([s1,∞),R). By Lemma 1, we see that y′(t) ≥ 0 for t ≥ s1. So
y(t) ≥ A(s1, t)y(s1) for t ≥ s1, from which it follows that
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 9
1240 SHAO YUAN HUANG, SUI SUN CHENG
v(t) = y(t) +A(s1, t)z(s1) ≥ A(s1, t) (y(s1) + z(s1)) = A(s1, t)x(s1) > 0 (26)
for t ≥ s1. We note that
v′(t) = y′(t) ≥ 0, (27)
and by (22)
x(t) = y(t) + z(t) ≥ y(t) +A(s1, t)z(s1) = v(t) (28)
for t ≥ s1. In view of (A3), there exists T > s1 such that rT > s1. By (28), then x(g(t)) ≥ v(g(t)) >
> 0 and
F (t, x(g(t))) ≥ F (t, v(g(t)))
for t ≥ T. By (27) and (28), we see that(
r(t)v′(t)
)′
+ F (t, v(g(t))) ≤
(
r(t)y′(t)
)′
+ F (t, x(g(t))) = 0, a.e. t ≥ T.
We observe that v(t+k ) = akv(tk) and v′(t+k ) = bkv
′(tk) for tk ≥ T. In view of (26), the function v
is an eventually positive solution of system(
r(t)u′(t)
)′
+ F (t, u(g(t))) ≤ 0, t ∈ [0,∞)\Υ,
u(t+k ) = aku(tk), k ∈ N, (29)
u′(t+k ) = bku
′(tk), k ∈ N,
such that v′(t) ≥ 0 for t ≥ T. By Lemma 2, the system (23) – (25) has an eventually positive solution
u such that v(t) ≥ u(t) and u′(t) ≥ 0 eventually. Assume that x is bounded and (A7) holds. By (22),
we note that the function z is bounded. Then the function v is bounded. So the function u is also
bounded.
Lastly, we assume that x is an eventually negative solution of (2) – (4). Let G(t, x) = −F (t,−x)
for x ∈ R. Let x̃(t) = −x(t) and z̃(t) = −z(t) for sufficiently large t. Then x̃ is an eventually
positive solution (
r(t)x′(t)
)′
+G(t, x(g(t))) = −e(t), t ∈ [0,∞)\Υ,
x(t+k ) = akx(tk), k ∈ N,
x′(t+k ) = bkx
′(tk), k ∈ N.
By (22), we note that
z̃(s̃n) = −z(s̃n) = − sup
{
z(t)
A(s̃n, t)
: t ≥ s̃n
}
= inf
{
z̃(t)
A(s̃n, t)
: t ≥ s̃n
}
,
for n ∈ N. By the former case, we see that the system(
r(t)u′(t)
)′
+G(t, u(g(t))) = 0, t ∈ [0,∞)\Υ,
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COMPARISON THEOREMS AND NECESSARY/SUFFICIENT CONDITIONS FOR EXISTENCE . . . 1241
u(t+k ) = aku(tk), k ∈ N,
u′(t+k ) = bku
′(tk), k ∈ N,
has an eventually positive solution ũ(t) such that ũ′(t) ≥ 0 eventually. Furthermore, ũ is bounded if
x̃ is bounded and (A7) holds. Let u(t) = −ũ(t) for sufficiently large t. So the system (23) – (25) has
an eventually negative solution u such that u′(t) ≤ 0 eventually, and u is bounded if x is bounded
and (A7) holds.
Theorem 2 is proved.
Theorem 3. Assume that (A1) – (A7) hold and the solution z of (5) satisfies limt→∞ z(t) = 0.
If the system (23) – (25) has a nonoscillatory solution u with u(t)u′(t) ≥ 0 eventually, then the
system (2) – (4) has a nonoscillatory solution x.
Proof. Assume that u is an eventually positive solution. Without loss of generality, we may
assume that u(t) > 0 and u′(t) ≥ 0 for t ≥ r0. Then
u(T ) ≥ A(0, T )u(0) ≥ mu(0) for any T ≥ 0. (30)
Since limt→∞ z(t) = 0, there exists T > 0 such that |z(t)| < m2u(0)/3 for t ≥ T. By the same
reasoning for obtaining the inequality (21), we have
u(t) ≥ A(T, t)
u(T ) +
t∫
T
1
r(s)
∞∫
s
F (v, u(g(v)))
B(s, v)
dvds
≥ m2u(0) (31)
for t ≥ 0. Let
X3 =
{
δ ∈ PC ([T,∞),R) : m2u(0)
3
≤ δ(t) ≤ u(t) for t ≥ T
}
.
Clearly, X3 is nonempty because u ∈ X3. We impose in X3 the same order relation imposed in X1.
Then X3 is a complete lattice. We define an operator H3 on X3 by
H3(δ)(t) = A(T, t)
2u(T )
3
+ z(t) +
t∫
T
A(s, t)
r(s)
∞∫
s
F (v, wu(δ)(v))
B(s, v)
dvds
for t ≥ T. In view of (30),
H3(δ)(t) ≥ m
2u(T )
3
− |z(t)| ≥ m2u(0)
3
, t ≥ T,
for any δ ∈ X3. Given δ1, δ2 ∈ X3 with δ1 ≤ δ2. We note that
0 ≤ wu(δ1)(t) ≤ wu(δ2)(t) ≤ u(g(t))
for t ≥ T. By assumption, (30) and (31), we have
H3(δ1)(t) ≤ H3(δ2)(t) ≤
≤ A(T, t)
2u(T )
3
+ |z(t)|+
t∫
T
A(s, t)
r(s)
∞∫
s
F (v, u(g(v)))
B(s, v)
dvds ≤
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1242 SHAO YUAN HUANG, SUI SUN CHENG
≤ A(T, t)
2u(T )
3
+m2u(0)
3
+
t∫
T
A(s, t)
r(s)
∞∫
s
F (v, u(g(v)))
B(s, v)
dvds <
< A(T, t)
2u(T )
3
+m
u(T )
3
+
t∫
T
A(s, t)
r(s)
∞∫
s
F (v, u(g(v)))
B(s, v)
dvds ≤ u(t)
for t ≥ T. So H3(X3) ⊆ X3 and H3 is increasing on X3. By the Knaster – Tarski fixed point theorem,
there exists θ3 ∈ X3 such that H3(θ3) = θ3. Let
x(t) =
θ3(t), t > T,
u(t), T ≥ t ≥ rT .
Let T1 > 0 such that rT1 > T. We note that x(g(t)) = wu(θ3)(t) for t ≥ T1. In view of H3(θ3) = θ3,
we have
x′(t) = z′(t) +
B(T2, t)
r(t)
∞∫
t
F (s, x(g(s)))
B(T2, s)
ds
from which it follows that x′ ∈ PC ′([T1,∞),R) and(
r(t)x′(t)
)′
+ F (t, x(g(t))) = e(t), a.e. t ≥ T1.
For any tk ≥ T2, by H3(θ3) = θ3 and (15),
x(t+k ) = akx(tk) and x′(t+k ) = bkx
′(tk).
Then x is a positive solution of system (2) – (4) on [T1,∞).
Theorem 3 is proved.
We remark that in the proof of Theorem 3, we see that in the condition (A7), we only need
“A(s, t) ≥ m for t ≥ s ≥ 0”.
The following result offers a necessary and sufficient oscillation theorem for (2) – (4).
Corollary 2. Assume that (A1) – (A7) and (6) hold, and the solution z of (5) is oscillatory
and limt→∞ z(t) = 0. Then the system (2) – (4) has a nonoscillatory solution x if, and only if,
system (23) – (25) has a nonoscillatory solution u.
Proof. We first show that there exist two sequences {sn}n∈N and {s̃n}n∈N such that (22) holds.
Let z̃(t) = z(t)/A(τ, t) for t ≥ τ. Clearly, z̃(t) is continuous on [τ,∞) and is oscillatory. By (A7)
and limt→∞ z(t) = 0, we see that limt→∞ z̃(t) = 0. In view of oscillation, there exists s1 > τ such
that z̃(s1) ≤ z̃(t) for t ≥ s1, which implies that
z(s1) ≤
z(t)
A(s1, t)
for t ≥ s1.
There exists s2 > s1 such that z̃(s2) ≤ z̃(t) for t ≥ s1, which implies that
z(s2) ≤
z(t)
A(s2, t)
for t ≥ s2.
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COMPARISON THEOREMS AND NECESSARY/SUFFICIENT CONDITIONS FOR EXISTENCE . . . 1243
By induction, we can take sequence {sn}n∈N such that (22) holds. Similarly, we can also take the
sequence {s̃n}n∈N such that (22) holds. By Theorem 2, the necessary condition holds. Conversely,
assume that u is nonoscillatory solution of system (23) – (25). By Lemma 1, we see that u(t)u′(t) ≥ 0
eventually. By Theorem 3, the sufficient condition holds.
Corollary 2 is proved.
We have another necessary and sufficient condition for the existence of bounded nonoscillatory
solutions of (2) – (4).
Corollary 3. Assume that (A1) – (A7) and (6) hold and F (t, µ) = p(t)f(µ) where p ∈ ALC([0,
∞), [0,∞)) and f is nondecreasing on R with µf(µ) > 0 for µ 6= 0. Assume that the solution z of
(5) is oscillatory, and there exist two sequences {sn}n∈N and {s̃n}n∈N such that (22) hold. Then the
system (2) – (4) has a bounded and nonoscillatory solution if, and only if,
∞∫
ε
1
r(s)
∞∫
s
p(v)
B(s, v)
dvds <∞ (32)
for some ε ≥ 0.
Proof. Assume that (32) holds. Clearly, (10) and (16) holds. By (A7) and (22), we note that the
function z is bounded. By Theorem 1 and Corollary 1, the necessary condition holds. Conversely, we
see that by Theorem 2, the system (23) – (25) has a nonoscillatory solution u such that u(t)u′(t) ≥ 0
eventually, and u is bounded. Assume that u is eventually positive with u′(t) ≥ 0 eventually. Then
there exists T > 0 such that u(t) > 0 and u′(t) ≥ 0 for t ≥ T. We observe that
Mu ≥ u(t) ≥ A(T, t)u(T ) ≥ mu(T ) > 0
for t ≥ T and some Mu > 0. So the function u has a positive lower bound. Since f is nondecreasing
on R, there is mf > 0 such that f(u(t)) ≥ mf for t ≥ T. We observe that(
r(t)u′(t)
u(t)
)′
=
(
r(t)u′(t)
u(t)
)′
≤ −p(t)f(u(g(t)))
u(t)
≤ −
mf
Mu
p(t)
for t ≥ T. We divide the above inequality by B(T, t)/A(T, t) and then integrate from t to d where
d ≥ t ≥ T. We have
A(T, d)r(d)u′(d)
B(T, d)u(d)
− A(T, t)r(t)u′(t)
B(T, t)u(t)
≤ −
mf
Mu
d∫
t
A(T, s)p(s)
B(T, s)
ds
for d ≥ t ≥ T. Since u′(d) > 0 and d is arbitrary, we see that
u′(t)
u(t)
≥
mf
Mur(t)
∞∫
t
A(t, s)p(s)
B(t, s)
ds
for t ≥ T, from which it follows that
u′(t) ≥ u′(t)mu(T )
u(t)
≥
m2u(T )mf
Mu
1
r(t)
∞∫
t
p(s)
B(t, s)
ds.
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1244 SHAO YUAN HUANG, SUI SUN CHENG
We divide the above inequality by A(T, t), and then integrate from T to t. Then
u(t) ≥ A(T, t)u(T ) +
m2u(T )mf
Mu
t∫
T
A(s, t)
r(s)
∞∫
t
p(s)
B(t, s)
ds ≥
≥ m
u(T ) +
mu(T )mf
Mu
t∫
T
1
r(s)
∞∫
t
p(s)
B(t, s)
ds
(33)
for t ≥ T. Since u is bounded, we may easily see that (32) holds. Assume that u is eventually
negative with u′(t) ≤ 0 eventually. Then −u is an eventually positive solution of system(
r(t)u′(t)
)′
+ p(t)f̃(u(g(t))) = 0, t ∈ [0,∞)\Υ, (34)
u(t+k ) = aku(tk), k ∈ N, (35)
u′(t+k ) = bku
′(tk), k ∈ N, (36)
where f̃(µ) = −f(−µ) for µ ∈ R. Similarly, the function −u has a positive lower bound. We note
that the function f̃ satisfies all assumption of f. So the condition (32) holds.
Corollary 3 is proved.
Theorem 4. Assume that the hypotheses of Corollary 3 hold except for the condition (A7), and
that g′(t) > 0 for t ≥ 0, ak ≥ 1 for k ∈ N, and
∞∫
ε
1
f(µ)
dµ <∞ for any ε > 0. (37)
If the system (2) – (4) has a nonoscillatory solution, then (32) holds.
Proof. Since the system (2) – (4) has a nonoscillatory solution, by Theorem 2, we see that sys-
tem (23) – (25) has a nonoscillatory solution u with u(t)u′(t) ≥ 0 eventually. Assume that u is
eventually positive with u′(t) ≥ 0 eventually. There exists T > 0 such that u(t) > 0 and u′(t) ≥ 0
for t ≥ tT . For any T1 > t ≥ T, we integrate (23) from t to T1, and we obtain
r(T1)u
′(T1)
B(T, T1)
− r(t)u′(t)
B(T, t)
+
T1∫
t
p(s)f(u(g(s)))
B(T, s)
ds = 0. (38)
Since T1 is arbitrary, by (38), we see that
r(t)u′(t)
B(T, t)
≥
∞∫
t
p(s)f(u(g(s)))
B(T, s)
ds, t ≥ T. (39)
Since g′(t) > 0 for t ≥ 0 and ak ≥ 1 for k ∈ N, we may see that g(s) ≥ g(t) and A(t, s) ≥ 1 for
s ≥ t ≥ T, from which it follows that
u(g(s)) ≥ A(g(t), g(s))u(g(t)) ≥ u(g(t)) (40)
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COMPARISON THEOREMS AND NECESSARY/SUFFICIENT CONDITIONS FOR EXISTENCE . . . 1245
for s ≥ t ≥ T. Since f is nondecreasing on R, we may further see that
f(u(g(s))) ≥ f(u(g(t))) > 0 (41)
for s ≥ t ≥ T. We divided (39) by f(u(g(t))). Then
u′(g(t))
f(u(g(t)))
≥ 1
r(g(t))
∞∫
g(t)
p(s)f(u(g(s)))
B(g(t), s)f(u(g(t)))
ds,
from which it follows that, by (41),
u′(g(t))
f(u(g(t)))
≥ 1
r(g(t))
∞∫
g(t)
p(s)
B(g(t), s)
ds (42)
for t ≥ T. We multiply (42) by g′(t), and then integrate from T to∞. We obtain
∞∫
T
(u(g(s)))′
f(u(g(s)))
ds ≥
∞∫
T
g′(s)
r(g(s))
∞∫
g(s)
p(v)
B(g(s), v)
dv
ds =
∞∫
g(T )
1
r(s)
∞∫
s
p(v)
B(s, v)
dv
ds. (43)
By (37), (40) and (43), we see that
∞∫
g(T )
1
r(s)
∞∫
s
p(v)
B(s, v)
dv
ds ≤
∞∫
T
(u(g(s)))′
f(u(g(s)))
ds ≤
∞∫
u(g(T ))
1
f(µ)
dµ <∞.
So (32) holds. Assume that u(t) is eventually negative. Let f̃(µ) = −f(−µ) for µ ∈ R. By (37),
we have
∞∫
ε
1
f̃(µ)
dµ <∞ for any ε > 0.
We note that−u(t) is eventually positive solution of system (34) – (36). Similarly, we may verify (32).
Theorem 4 is proved.
Recall now the equation (1) under the condition η > 1. For the ease of discussion, we state the
result in [1].
Corollary 4 [1]. Let η > 1 be given and let e, p and g be continuous functions on [0,∞) such
that limt→∞ g(t) = ∞, p(t) ≥ 0, g(t) ≤ t and g′(t) ≥ α > 0 for t ≥ 0. Assume that there exists
a bounded function z(t) on [0,∞) such that z′′(t) = e(t) for all sufficient large t, and that z is
oscillatory and (22) holds. Then
∞∫
0
sp(s)ds <∞ (44)
if, and only if, the equation (1) has a nonoscillatory solution.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 9
1246 SHAO YUAN HUANG, SUI SUN CHENG
Because the equation (1) has no impulsive effects, ak = bk = 1 for all k. That is, A(s, t) =
= B(s, t) = 1 for t ≥ s ≥ 0. Let r(t) = 1 for t ≥ 0. Clearly, (6) holds and
∞∫
0
1
r(s)
∞∫
s
p(v)
B(s, v)
dvds =
∞∫
0
sp(s)ds.
So either condition (10) or (32) is equivalent to (44). We have the following conclusions.
(1) Assume that we replace the condition “g′(t) ≥ α > 0 for t ≥ 0” by g′(t) > 0 for t ≥ 0. By
Theorem 1 and 4, we can obtain the same Corollary in [1].
(2) Assume that we remove the condition “g′(t) ≥ α > 0 for t ≥ 0”. By Corollary 3, we can
see that (44) holds if, and only if, the equation (1) has a nonoscillatory and bounded solution. We
note that if the equation (1) has a nonoscillatory and unbounded solution, the condition (44) may not
be true. So this result is sharp without the condition “g′(t) > 0 for t ≥ 0.” We give an example to
illustrate it. Let ε1(t) = t1/3 (2 + sin t) for t ≥ 0. Clearly, there exists a > 0 such that ε1(t) ≤ t for
t ≥ a. Let
g(t) =
ε1(t), t ≥ a,
ε2(t), a > t ≥ 0,
where ε2 is a nonnegative and continuous function on [0, a] with ε2(t) ≤ t for 0 ≤ t ≤ a, and
ε1(a) = ε2(a). By simple computation, we can see that limt→∞ g(t) = ∞ and g(t) ≤ t for t ≥ 0,
and it is impossible that g′(t) > 0 for sufficiently large t. We consider a special equation
x′′(t) +
1
4t3/2g(t)
x2(g(t)) = 0, t ≥ 0. (45)
Then the function x(t) =
√
t is an eventually positive solution of (45) and is unbounded. But we can
see that
∞∫
a
t
4t3/2g(t)
dt =
∞∫
a
1
4t5/6 (2 + sin t)
dt ≥
∞∫
a
1
12t5/6
dt =∞.
Hence we have indeed made an improvement by avoiding the condition “g′(t) ≥ α > 0”.
3. Example. Assume that (A2) and (A3) hold. Let α ∈ R and the function
p(t) =
t
α if t > 0,
0 if t = 0.
Consider the Klein – Gordon equation (c.f. Example 2.6.3 in [5])
x′′(t) + p(t) |x(g(t))| exp(|x(g(t))|2) = e−t sin t, t ∈ [0,∞)\Υ, (46)
x(t+k ) = akx(tk), k ∈ N, (47)
x′(t+k ) = akx
′(tk), k ∈ N, (48)
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COMPARISON THEOREMS AND NECESSARY/SUFFICIENT CONDITIONS FOR EXISTENCE . . . 1247
where ak are positive constants for k ∈ N such that (A7) holds. We can easily check that
z(t) = A(0, t)
t∫
3π
2
s∫
3π
4
e−v sin v
A(0, v)
dvds, t ≥ 3π
2
,
is a solution of the system
z′′(t) = e−t sin t, t ∈ [0,∞)\Υ,
z(t+k ) = akz(tk), k ∈ N,
z′(t+k ) = akz
′(tk), k ∈ N.
Since
z(t) ≥ m
t∫
3π
2
s∫
3π
4
e−v sin vdvds ≥ m
(
e−t
2
cos t
)
and
z(t) ≤M
t∫
3π
2
s∫
3π
4
e−v sin vdvds = M
(
e−t
2
cos t
)
for t ≥ 3π/2, we may see that z(t) is oscillatory and limt→∞ z(t) = 0. Before stating the following
conclusions, recall that a function ϕ defined for all sufficiently large t is oscillatory if ϕ is neither
eventually positive nor eventually negative.
(1) It is easy to check that all hypotheses of Corollary 3 are satisfied. We first note that α < −2
if, and only if,
∞∫
1
∞∫
s
vα
A(s, v)
dvds <∞.
By Corollary 3, we can see that α < −2 if, and only if, the system (46) – (48) has a nonoscillatory
solution x which is bounded.
(2) It is easy to check that all hypotheses of Corollary 2 are satisfied. Assume that m ≥ 1 and
α ≥ −1. Then
∞∫
1
p(t)
A(0, t)
dt =
∞∫
1
tα
A(0, t)
dt ≥
∞∫
1
tα
M
dt =∞.
Since
∞∑
i=1
∏
1≤tj<ti
1
aj
ti∫
ti−1
p(t)dt =
∞∫
0
tα
A(0, t)
dt =∞.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 9
1248 SHAO YUAN HUANG, SUI SUN CHENG
By Theorem 1 in [4], then every solution of system
x′′(t) + p(t) |x(g(t))| exp(|x(g(t))|2) = 0, t ∈ [0,∞)\Υ, (49)
x(t+k ) = akx(tk), k ∈ N, (50)
x′(t+k ) = akx
′(tk), k ∈ N, (51)
is oscillatory. By Corollary 3, we can further see that every solution of system (46) – (48) is oscillatory.
(3) Assume that g′(t) > 0 for t ≥ 0, ak ≥ 1 for k ∈ N. Since
∞∫
ε
e−µ
µ
dµ ≤ 1
ε
∞∫
ε
e−µdµ <∞ for any ε > 0,
by Theorem 4, we see that if α ≥ −2, then every solution of system (46) – (48) is oscillatory.
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Received 16.09.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 9
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| id | umjimathkievua-article-2654 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:27:42Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/0b/8a5f0deac381ef3052e7249ff63ac10b.pdf |
| spelling | umjimathkievua-article-26542020-03-18T19:32:05Z Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations Теореми порiвняння та необхiднi/достатнi умови iснування неосциляцiйних розв’язкiв збурених iмпульсних диференцiальних рiвнянь iз запiзненням Cheng, Sui Sun Shao, Yuan Huang Ченг, Суй Сун Хуан, Шао Юань In 1997, A. H. Nasr provided necessary and sufficient conditions for the oscillation of the equation $$x''(t) + p(t) |x(g(t))|^{\eta} \text{sgn} (x(g(t))) = e(t),$$ where $\eta > 0$, $p$, and $g$ are continuous functions on $[0, \infty)$ such that $p(t) \geq 0,\;\; g(t) \leq t,\;\; g'(t) \geq \alpha > 0$, and $\lim_{t \rightarrow \infty} g(t) = \infty$ It is important to note that the condition $g'(t) \geq \alpha > 0$ is required. In this paper, we remove this restriction under the superlinear assumption $\eta > 0$. Infact, we can do even better by considering impulsive differential equations with delay and obtain necessary and sufficient conditions for the existence of nonoscillatory solutions and also a comparison theorem that enables us to apply known oscillation results for impulsive equations without forcing terms to yield oscillation criteria for our equations. У 1997 роцi, А. Х. Наср отримав необхiднi та достатнi осциляцiйнi умови для рiвняння $$x''(t) + p(t) |x(g(t))|^{\eta} \text{sgn} (x(g(t))) = e(t),$$ де $\eta > 0$, $p$ та $g$ — неперервнi функцiї на $[0, \infty)$ такi, що $p(t) \geq 0,\;\; g(t) \leq t,\;\; g'(t) \geq \alpha > 0$ та $\lim_{t \rightarrow \infty} g(t) = \infty$. Слiд зауважити, що необхiдною тут є умова $g'(t) \geq \alpha > 0$. У данiй статтi ми усуваємо це обмеження при суперлiнiйному припущеннi $\eta > 0$. Насправдi, можна отримати навiть кращий результат, розглядаючи iмпульснi диференцiальнi рiвняння з запiзненням, i встановити необхiднi та достатнi умови iснування неосциляцiйних розв’язкiв, а також теорему порiвняння, яка дає змогу застосувати вiдомi осциляцiйнi результати для iмпульсних рiвнянь без збурюючих членiв, щоб отримати осциляцiйнi критерiї для наших рiвнянь. Institute of Mathematics, NAS of Ukraine 2012-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2654 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 9 (2012); 1233-1248 Український математичний журнал; Том 64 № 9 (2012); 1233-1248 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2654/2067 https://umj.imath.kiev.ua/index.php/umj/article/view/2654/2068 Copyright (c) 2012 Cheng Sui Sun; Shao Yuan Huang |
| spellingShingle | Cheng, Sui Sun Shao, Yuan Huang Ченг, Суй Сун Хуан, Шао Юань Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations |
| title | Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations |
| title_alt | Теореми порiвняння та необхiднi/достатнi умови iснування неосциляцiйних розв’язкiв збурених iмпульсних диференцiальних рiвнянь iз запiзненням |
| title_full | Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations |
| title_fullStr | Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations |
| title_full_unstemmed | Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations |
| title_short | Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations |
| title_sort | comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2654 |
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