Solvability of inhomogeneous boundary-value problems for fourth-order differential equations
Some new oscillation criteria are established for the nonlinear damped differential equation $$(r(t)k_1 (x, x'))' + p (t) k_2 (x, x') x' + q (t) f (x (t)) = 0,\quad t \geq t_0.$$ The results obtained extend and improve some existing results in the literature.
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Institute of Mathematics, NAS of Ukraine
2011
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508777871048704 |
|---|---|
| author | Avci, H. Tunç, E. Авкі, Х. Тунс, Е. |
| author_facet | Avci, H. Tunç, E. Авкі, Х. Тунс, Е. |
| author_sort | Avci, H. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:36:55Z |
| description | Some new oscillation criteria are established for the nonlinear damped differential equation
$$(r(t)k_1 (x, x'))' + p (t) k_2 (x, x') x' + q (t) f (x (t)) = 0,\quad t \geq t_0.$$
The results obtained extend and improve some existing results in the literature. |
| first_indexed | 2026-03-24T02:30:36Z |
| format | Article |
| fulltext |
UDC 517.9
E. Tunç (Gaziosmanpas.a Univ., Turkey),
H. Avcı (Ondokuz Mayıs Univ., Turkey)
NEW OSCILLATION THEOREMS
FOR A CLASS OF SECOND-ORDER DAMPED
NONLINEAR DIFFERENTIAL EQUATIONS
НОВI ОСЦИЛЯЦIЙНI ТЕОРЕМИ
ДЛЯ ОДНОГО КЛАСУ ЗГАСАЮЧИХ НЕЛIНIЙНИХ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ ДРУГОГО ПОРЯДКУ
Some new oscillation criteria are established for the nonlinear damped differential equation(
r(t)k1
(
x, x′))′ + p (t) k2
(
x, x′)x′ + q (t) f (x (t)) = 0, t ≥ t0.
The results obtained extend and improve some existing results in the literature.
Встановлено деякi новi осциляцiйнi критерiї для згасаючого нелiнiйного рiвняння(
r(t)k1
(
x, x′))′ + p (t) k2
(
x, x′)x′ + q (t) f (x (t)) = 0, t ≥ t0.
Отриманi результати узагальнюють i посилюють деякi iснуючi результати.
1. Introduction. We are concerned with oscillation behavior of solutions of second-
order nonlinear differential equations with nonlinear damping of the form
(r(t)k1 (x, x
′))
′
+ p (t) k2 (x, x
′)x′ + q (t) f (x (t)) = 0, (1.1)
where t ≥ t0 > 0, r ∈ C1([t0,∞) ; (0,∞)), p, q ∈ C ([t0,∞) ;R), f ∈ C (R,R) ,
k1 ∈ C1
(
R2, R
)
and k2 ∈ C
(
R2, R
)
. Throughout the paper, it is assumed that
(a) p(t) ≥ 0 for all t ≥ t0;
(b) f(x)/x ≥ K for some constant K > 0 and all x ∈ R\ {0} ;
(c) q(t) ≥ 0 for all t ≥ t0 and q(t) 6≡ 0 on [t∗, 0) for any t∗ ≥ t0;
(d) k21(u, v) ≤ α1vk1(u, v) for some constant α1 > 0 and all (u, v) ∈ R2;
(e) uvk2(u, v) ≥ α2k
2
1(u, v) for some constant α2 > 0 and all (u, v) ∈ R2;
(e1) uvk2(u, v) ≥ α2uk1(u, v) for some constant α2 > 0 and all (u, v) ∈ R2.
We recall that a function x : [t0, t1) → (−∞,∞), t1 > t0, is called a solution of
equation (1.1) if x(t) satisfies equation (1.1) for all t ∈ [t0, t1). In the sequel, it will
be always assumed that solutions of equation (1.1) exist on [t0,∞). A solution x(t) of
equation (1.1) is called oscillatory if it has arbitrary large zeros, otherwise it is called
nonoscillatory. Eq. (1.1) is called oscillatory if all solutions are oscillatory.
In the relevant literature, till now, oscillation behaviors of solutions of linear and
non-linear second order differential equations have been the subject of intensive investi-
gations for many authors. For instance, one can refer to [1 – 39], as some related papers
or books on the subject.
The oscillation of Eq. (1.1) was first studied by Rogovchenko and Rogovchenko
in [1]. Afterward, under assumptions (a) – (e), Tiryaki and Zafer [21] established some
oscillation criteria for (1.1), which extend and improve the results in [1]. We also note
that the similar results for the differential equations those are near to (1) were established
before (see, for example [36]).
The motivation for the present work has been inspired basically by the paper of
[1, 21] and the works mentioned above. Our aim here is to improve the some results
c© E. TUNÇ, H. AVCI, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9 1263
1264 E. TUNÇ, H. AVCI
verified by [21] for the oscillation of solutions of Eq. (1.1) under assumptions (a) – (e).
On the other hand, under assumptions (a) – (d) and (e1) our results are new.
Let D0 = {(t, s) : t > s ≥ t0} and D = {(t, s) : t ≥ s ≥ t0} . We say that the func-
tion H = H(t, s) ∈ C (D, (−∞,∞)) belongs to the class P if
(i) H(t, t) = 0 for t ≥ t0, H(t, s) > 0 on D0;
(ii) H(t, s) has a continuous and nonpositive partial derivative on D0 with respect
to the second variable, and there is a function h ∈ C (D, [0,+∞)) such that
−∂H
∂s
(t, s) = h(t, s)
√
H(t, s) for all (t, s) ∈ D0.
2. Main results. The main results of this paper are the following theorems.
Theorem 1. Let assumptions (a) – (d) and (e1) hold. Further, suppose that there
exists a function g ∈ C1 ([t0,∞) ;R) such that, for some β ≥ 1 and for some H ∈ P,
lim sup
t→∞
1
H(t, t0)
t∫
t0
(
H(t, s)γ(s)− α1βr(s)v(s)
4
h2(t, s)
)
ds =∞, (2.1)
where
v(t) = exp
−2 t∫ (
g(s)
α1
− α2p(s)
2r(s)
)
ds
(2.2)
and
γ(t) = v(t)
(
Kq(t) +
r(t)g2(t)
α1
− α2p(t)g(t)− (r(t)g(t))
′
)
. (2.3)
Then Eq. (1.1) is oscillatory.
Proof. Let x(t) be a non-oscillatory solution of Eq. (1.1). Then there exists a T0 ≥
≥ t0 such that x(t) 6= 0 for all t ≥ T0 ≥ t0. For t ≥ T0, define a generalized Riccati
transformation by
w(t) = v(t)
[
r(t)k1 (x (t) , x
′ (t))
x (t)
+ r(t)g(t)
]
(2.4)
where v(t) is given by (2.2). Differentiating (2.4) and using (1.1), we have
w′(t) =
v′(t)
v(t)
w(t) + v(t)
[
−p(t)k2 (x (t) , x′ (t))x′ (t)
x(t)
−
−q(t)f(x(t))
x(t)
− r(t)k1 (x (t) , x
′ (t))x′(t)
x2 (t)
+ (r(t)g(t))
′
]
. (2.5)
Using (a) – (d) and (e1) in (2.5), we obtain
w′(t) ≤ v′(t)
v(t)
w(t)− α2p(t)v(t)k1 (x (t) , x
′ (t))
x(t)
−Kq(t)v(t)−
−r(t)v(t)k
2
1 (x (t) , x
′ (t))
α1x2(t)
+ v(t) (r(t)g(t))
′
=
=
v′(t)
v(t)
w(t)− α2p(t)v(t)
[
w(t)
v(t)r(t)
− g(t)
]
−Kq(t)v(t)−
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9
NEW OSCILLATION THEOREMS FOR A CLASS OF SECOND-ORDER DAMPED . . . 1265
−r(t)v(t)
α1
[
w(t)
v(t)r(t)
− g(t)
]2
+ v(t) (r(t)g(t))
′
=
=
[
−2g(t)
α1
+
α2p(t)
r(t)
]
w(t)− α2p(t)w(t)
r(t)
+ α2p(t)g(t)v(t)−Kq(t)v(t)−
− w2(t)
α1v(t)r(t)
+
2g(t)w(t)
α1
− r(t)g2(t)v(t)
α1
+ v(t) (r(t)g(t))
′
. (2.6)
So, (2.6) yields, for all t ≥ T0,
w′(t) ≤ −γ(t)− w2(t)
α1v(t)r(t)
(2.7)
where γ(t) is defined by (2.3). Multiplying both sides of (2.7) by H(t, s) and integrating
from T to t, we have for some β ≥ 1 and for all t ≥ T ≥ T0,
t∫
T
H(t, s)γ(s)ds ≤ −
t∫
T
H(t, s)w′(s)ds−
t∫
T
H(t, s)
w2(s)
α1r(s)v(s)
ds =
= H(t, T )w(T )−
t∫
T
[
−∂H(t, s)
∂s
]
w(s)ds−
t∫
T
H(t, s)
w2(s)
α1r(s)v(s)
ds =
= H(t, T )w(T )−
t∫
T
[
h(t, s)
√
H(t, s)w(s) +H(t, s)
w2(s)
α1r(s)v(s)
]
ds =
= H(t, T )w(T )−
t∫
T
(√
H(t, s)
βα1r(s)v(s)
w(s) +
1
2
√
βα1r(s)v(s)h(t, s)
)2
ds+
+
βα1
4
t∫
T
r(s)v(s)h2(t, s)ds−
t∫
T
(β − 1)H(t, s)
βα1r(s)v(s)
w2(s)ds.
Hence, for all t ≥ T ≥ T0,
t∫
T
(
H(t, s)γ(s)− α1β
4
r(s)v(s)h2(t, s)
)
ds ≤
≤ H(t, T )w(T )−
t∫
T
(β − 1)H(t, s)
βα1r(s)v(s)
w2(s)ds−
−
t∫
T
(√
H(t, s)
βα1r(s)v(s)
w(s) +
1
2
√
βα1r(s)v(s)h(t, s)
)2
ds. (2.8)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9
1266 E. TUNÇ, H. AVCI
So, for every t ≥ T0,
t∫
T0
(
H(t, s)γ(s)− α1β
4
r(s)v(s)h2(t, s)
)
ds ≤
≤ H(t, T0)w(T0) ≤ H(t, T0) |w(T0)| ≤ H(t, t0) |w(T0)| .
Therefore,
t∫
t0
(
H(t, s)γ(s)− α1β
4
r(s)v(s)h2(t, s)
)
ds =
=
T0∫
t0
(
H(t, s)γ(s)− α1β
4
r(s)v(s)h2(t, s)
)
ds+
+
t∫
T0
(
H(t, s)γ(s)− α1β
4
r(s)v(s)h2(t, s)
)
ds ≤
≤
T0∫
t0
H(t, s) |γ(s)| ds+H(t, t0) |w(T0)| ≤
≤ H(t, t0)
T0∫
t0
|γ(s)| ds+ |w(T0)|
for all t ≥ T0. This gives
t∫
t0
(
H(t, s)γ(s)− α1β
4
r(s)v(s)h2(t, s)
)
ds ≤ H(t, t0)
T0∫
t0
|γ(s)| ds+ |w(T0)|
.
(2.9)
It follows from (2.9) that
lim sup
t→∞
1
H(t, t0)
t∫
t0
(
H(t, s)γ(s)− α1β
4
r(s)v(s)h2(t, s)
)
ds ≤
≤
T0∫
t0
|γ(s)| ds+ |w(T0)|
< +∞
which contradicts (2.1). Therefore, Eq. (1.1) is oscillatory.
Under a modification of the hypotheses of Theorem 1, we can obtain the following
result.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9
NEW OSCILLATION THEOREMS FOR A CLASS OF SECOND-ORDER DAMPED . . . 1267
Corollary 1. In Theorem 1, if condition (2.1) is replaced by the conditions
lim sup
t→∞
1
H(t, t0)
t∫
t0
H(t, s)γ(s)ds =∞
and
lim sup
t→∞
1
H(t, t0)
t∫
t0
r(s)v(s)h2(t, s)ds <∞,
then (1.1) is oscillatory.
Theorem 2. Let assumptions (a) – (d) and (e1) be fulfilled. Suppose that there
exists function H ∈ P and
0 < inf
s≥t0
[
lim inf
t→∞
H(t, s)
H(t, t0)
]
≤ ∞. (2.10)
Assume that there exist functions g ∈ C1 ([t0,∞) ;R) and b ∈ C ([t0,∞) ;R) such that,
for some β > 1, all t > t0 and T ≥ t0,
lim sup
t→∞
1
H(t, T )
t∫
T
(
H(t, s)γ(s)− α1β
4
r(s)v(s)h2(t, s)
)
ds ≥ b(T ), (2.11)
where γ(s), v(t) are as in Theorem 1. If
lim sup
t→∞
t∫
t0
b2+(s)
r(s)v(s)
ds =∞, (2.12)
where b+(t) = max{b(t), 0}. Eq. (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of (1.1). We may assume that x(t) 6= 0
for some t ≥ T0 ≥ t0. Define w(t) as in (2.4). Then, following the proof of Theorem 1,
we obtain (2.8). Further, it follows that
1
H(t, T )
t∫
T
(
H(t, s)γ(s)− α1β
4
r(s)v(s)h2(t, s)
)
ds ≤
≤ w(T )− 1
H(t, T )
t∫
T
(β − 1)H(t, s)
βα1r(s)v(s)
w2(s)ds−
− 1
H(t, T )
t∫
T
(√
H(t, s)
βα1r(s)v(s)
w(s) +
1
2
√
βα1r(s)v(s)h(t, s)
)2
ds ≤
≤ w(T )− 1
H(t, T )
t∫
T
(β − 1)H(t, s)
βα1r(s)v(s)
w2(s)ds.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9
1268 E. TUNÇ, H. AVCI
Hence, for t > T ≥ T0,
lim sup
t→∞
1
H(t, T )
t∫
T
(
H(t, s)γ(s)− α1β
4
r(s)v(s)h2(t, s)
)
ds ≤
≤ w(T )− lim inf
t→∞
1
H(t, T )
t∫
T
(β − 1)H(t, s)
βα1r(s)v(s)
w2(s)ds.
Thus, by (2.11) we have
w(T ) ≥ b(T ) + lim inf
t→∞
1
H(t, T )
t∫
T
(β − 1)H(t, s)
βα1r(s)v(s)
w2(s)ds,
for all T ≥ T0 and for any β > 1. This implies that
w(T ) ≥ b(T ) for all T ≥ T0 (2.13)
and
lim inf
t→∞
1
H(t, T0)
t∫
T0
H(t, s)
r(s)v(s)
w2(s)ds ≤ βα1
(β − 1)
(w(T0)− b(T0)) <∞. (2.14)
Now, we calim that
∞∫
T0
w2(s)
r(s)v(s)
ds <∞. (2.15)
Suppose to the contrary that
∞∫
T0
w2(s)
r(s)v(s)
ds =∞. (2.16)
By (2.10), there exists a positive constant δ such that
inf
s≥t0
[
lim inf
t→∞
H(t, s)
H(t, t0)
]
> δ > 0. (2.17)
From (2.17),
lim inf
t→∞
H(t, s)
H(t, t0)
> δ > 0,
and there exists a T2 ≥ T1 such that H(t, T1)/H(t, t0) ≥ δ, for all t ≥ T2. On the
other hand, by (2.16), for any positive number κ, there exist a T1 > T0, such that, for
all t ≥ T1,
t∫
T0
w2(s)
r(s)v(s)
ds ≥ κ
δ
.
Using integration by parts, we conclude that, for all t ≥ T1,
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9
NEW OSCILLATION THEOREMS FOR A CLASS OF SECOND-ORDER DAMPED . . . 1269
1
H(t, T0)
t∫
T0
H(t, s)
r(s)v(s)
w2(s)ds =
1
H(t, T0)
t∫
T0
[
−∂H(t, s)
∂s
] s∫
T0
w2(τ)
r(τ)v(τ)
dτ
ds ≥
≥ κ
δ
1
H(t, T0)
t∫
T1
[
−∂H(t, s)
∂s
]
ds =
κ
δ
H(t, T1)
H(t, T0)
. (2.18)
Hence, we have from (2.18)
1
H(t, T0)
t∫
T0
H(t, s)
r(s)v(s)
w2(s)ds ≥ κ for all t ≥ T2.
Since κ is an arbitrary positive constant,
lim inf
t→∞
1
H(t, T0)
t∫
T0
H(t, s)
r(s)v(s)
w2(s)ds = +∞,
which contradicts assumption (2.14). Therefore we proved that (2.16) fails, so (2.15)
holds true. Then, it follows from (2.13) that b2+(T ) ≤ w2(T ) for all T ≥ T0, and
∞∫
T0
b2+(s)
r(s)v(s)
ds ≤
∞∫
T0
w2(s)
r(s)v(s)
ds < +∞,
which contradicts (2.12). Hence, Eq. (1.1) is oscillatory.
Theorem 3. The conclusion of Theorem 2 remains valid, if assumptions (2.11) is
replaced by
lim inf
t→∞
1
H(t, T )
t∫
T
(
H(t, s)γ(s)− α1β
4
r(s)v(s)h2(t, s)
)
ds ≥ b(T )
for all T ≥ t0 and for some β > 1.
Proof. Due to the fact that
b(T ) ≤ lim inf
t→∞
1
H(t, T )
t∫
T
(
H(t, s)γ(s)− α1β
4
r(s)v(s)h2(t, s)
)
ds ≤
≤ lim sup
t→∞
1
H(t, T )
t∫
T
(
H(t, s)γ(s)− α1β
4
r(s)v(s)h2(t, s)
)
ds,
the conclusion follows immediately from Theorem 2.
From now on, we shall consider the oscillation for (1.1) under assumptions (a) – (e).
Theorem 4. Let assumptions (a) – (e) hold. Suppose that there exists a function
g ∈ C1 ([t0,∞) ;R) such that, for some β ≥ 1 and for some H ∈ P,
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9
1270 E. TUNÇ, H. AVCI
lim sup
t→∞
1
H(t, t0)
t∫
t0
(
H(t, s)γ1(s)−
βα1
4(α1α2p(s) + r(s))
r2(s)v(s)h2(t, s)
)
ds =∞,
(2.19)
where
v(t) = exp
−2 t∫ (
g(s)
α1
+
α2p(s)g(s)
r(s)
)
ds
, (2.20)
γ1(t) = v(t)
(
Kq(t) +
r(t)g2(t)
α1
+ α2p(t)g
2(t)− (r(t)g(t))
′
)
. (2.21)
Then Eq. (1.1) is oscillatory.
Proof. Let x(t) be a non-oscillatory solution of the differential equation (1.1). Then
there exists a T0 ≥ t0 such that x(t) 6= 0 for all t ≥ T0. Define
w(t) = v(t)
[
r(t)k1 (x (t) , x
′ (t))
x (t)
+ r(t)g(t)
]
for all t ≥ T0,
where v(t) is given by (2.20). This and (1.1) imply
w′(t) =
v′(t)
v(t)
w(t) + v(t)
[
−p(t)k2 (x (t) , x′ (t))x′ (t)
x(t)
−
−q(t)f(x(t))
x(t)
− r(t)k1 (x (t) , x
′ (t))x′(t)
x2 (t)
+ (r(t)g(t))
′
]
.
Taking into account (a) – (e), we conclude that for all t ≥ T0,
w′(t) ≤ v′(t)
v(t)
w(t)− α2p(t)v(t)k
2
1 (x (t) , x
′(t))
x2(t)
−Kq(t)v(t)−
−r(t)v(t)k
2
1 (x (t) , x
′ (t))
α1x2(t)
+ v(t) (r(t)g(t))
′
=
=
v′(t)
v(t)
w(t)− α2p(t)v(t)
[
w(t)
v(t)r(t)
− g(t)
]2
−Kq(t)v(t)−
−r(t)v(t)
α1
[
w(t)
v(t)r(t)
− g(t)
]2
+ v(t) (r(t)g(t))
′
=
=
[
−2g(t)
α1
− 2α2p(t)g(t)
r(t)
]
w(t)− α2p(t)w
2(t)
v(t)r2(t)
+
2α2p(t)g(t)w(t)
r(t)
−
−α2p(t)g
2(t)v(t)−Kq(t)v(t)− w2(t)
α1v(t)r(t)
+
+
2g(t)w(t)
α1
− r(t)g2(t)v(t)
α1
+ v(t) (r(t)g(t))
′
which yields
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NEW OSCILLATION THEOREMS FOR A CLASS OF SECOND-ORDER DAMPED . . . 1271
w′(t) ≤ −γ1(t)−
(α1α2p(t) + r(t))w2(t)
α1v(t)r2(t)
, (2.22)
where γ1(t) is defined by (2.21). Multiplying both sides of (2.22) by H(t, s) and inte-
grating from T to t, we have for some β ≥ 1 and for all t ≥ T ≥ T0,
t∫
T
H(t, s)γ1(s)ds ≤ −
t∫
T
H(t, s)w′(s)ds−
t∫
T
H(t, s)
(α1α2p(s) + r(s))w2(s)
α1r2(s)v(s)
ds =
= H(t, T )w(T )−
t∫
T
[
−∂H(t, s)
∂s
]
w(s)ds−
t∫
T
H(t, s)
(α1α2p(s) + r(s))w2(s)
α1r2(s)v(s)
ds =
= H(t, T )w(T )−
t∫
T
[
h(t, s)
√
H(t, s)w(s) +H(t, s)
(α1α2p(s) + r(s))w2(s)
α1r2(s)v(s)
]
ds =
= H(t, T )w(T )−
t∫
T
(√
H(t, s) (α1α2p(s) + r(s))
βα1r2(s)v(s)
w(s)+
+
1
2
√
βα1r2(s)v(s)
(α1α2p(s) + r(s))
h(t, s)
)2
ds+
βα1
4
t∫
T
r2(s)v(s)
(α1α2p(s) + r(s))
h2(t, s)ds−
−
t∫
T
(β − 1)H(t, s) (α1α2p(s) + r(s))
βα1r2(s)v(s)
w2(s)ds.
Thus, for all t ≥ T ≥ T0, we obtain that
t∫
T
(
H(t, s)γ1(s)−
βα1r
2(s)v(s)
4 (α1α2p(s) + r(s))
h2(t, s)
)
ds ≤ H(t, T )w(T )−
−
t∫
T
(β − 1)H(t, s) (α1α2p(s) + r(s))
βα1r2(s)v(s)
w2(s)ds−
−
t∫
T
(√
H(t, s) (α1α2p(s) + r(s))
βα1r2(s)v(s)
w(s) +
1
2
√
βα1r2(s)v(s)
(α1α2p(s) + r(s))
h(t, s)
)2
ds.
(2.23)
Following the same lines as in the proof of Theorem 1, we conclude that
lim sup
t→∞
1
H(t, t0)
t∫
t0
(
H(t, s)γ1(s)−
βα1r
2(s)v(s)
4 (α1α2p(s) + r(s))
h2(t, s)
)
ds ≤
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9
1272 E. TUNÇ, H. AVCI
≤
T0∫
t0
|γ1(s)| ds+ |w(T0)| < +∞,
which contradicts assumption (2.19). Therefore, we have proved that all solutions of
Eq. (1.1) are oscillatory.
Corollary 2. The conclusion of Theorem 4 remains intact if assumption (2.19) is
replaced with the following conditions:
lim sup
t→∞
1
H(t, t0)
t∫
t0
H(t, s)γ1(s)ds =∞
and
lim sup
t→∞
1
H(t, t0)
t∫
t0
r2(s)v(s)
α1α2p(s) + r(s)
h2(t, s)ds <∞.
Theorem 5. Let assumptions (a) – (e) hold. Suppose that there exists function H ∈
∈ P and
0 < inf
s≥t0
[
lim inf
t→∞
H(t, s)
H(t, t0)
]
≤ ∞. (2.24)
Assume that there exist functions g ∈ C1 ([t0,∞) ;R) and b ∈ C ([t0,∞) ;R) such that,
for all t > t0, all T ≥ t0, and for some β > 1,
lim sup
t→∞
1
H(t, T )
t∫
T
(
H(t, s)γ1(s)−
βα1r
2(s)v(s)
4 (α1α2p(s) + r(s))
h2(t, s)
)
ds ≥ b(T ),
(2.25)
where γ1(s), v(t) are as in Theorem 4, and furthermore suppose that
lim sup
t→∞
t∫
t0
(α1α2p(s) + r(s)) b2+(s)
r2(s)v(s)
ds =∞, (2.26)
where b+(t) = max {b(t), 0} . Then Eq. (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of (1.1). We may assume that x(t) 6= 0
for some t ≥ T0 ≥ t0. Define w(t) as in (2.4). As in the proof of Theorem 4, we can
obtain (2.23). Then,
1
H(t, T )
t∫
T
(
H(t, s)γ1(s)−
βα1r
2(s)v(s)
4 (α1α2p(s) + r(s))
h2(t, s)
)
ds ≤ w(T )−
− 1
H(t, T )
t∫
T
(β − 1)H(t, s) (α1α2p(s) + r(s))
βα1r2(s)v(s)
w2(s)ds−
− 1
H(t, T )
t∫
T
(√
H(t, s) (α1α2p(s) + r(s))
βα1r2(s)v(s)
w(s)+
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NEW OSCILLATION THEOREMS FOR A CLASS OF SECOND-ORDER DAMPED . . . 1273
+
1
2
√
βα1r2(s)v(s)
(α1α2p(s) + r(s))
h(t, s)
)2
ds ≤
≤ w(T )− 1
H(t, T )
t∫
T
(β − 1)H(t, s) (α1α2p(s) + r(s))
βα1r2(s)v(s)
w2(s)ds.
Consequently,
lim sup
t→∞
1
H(t, T )
t∫
T
(
H(t, s)γ1(s)−
βα1r
2(s)v(s)
4 (α1α2p(s) + r(s))
h2(t, s)
)
ds ≤
≤ w(T )− lim inf
t→∞
1
H(t, T )
t∫
T
(β − 1)H(t, s) (α1α2p(s) + r(s))
βα1r2(s)v(s)
w2(s)ds
for t > T ≥ T0. It follows from (2.25)
w(T ) ≥ b(T ) + lim inf
t→∞
1
H(t, T )
t∫
T
(β − 1)H(t, s) (α1α2p(s) + r(s))
βα1r2(s)v(s)
w2(s)ds
for all T ≥ T0 and for any β > 1. This implies that,
w(T ) ≥ b(T ) for all T ≥ T0 (2.27)
and
lim inf
t→∞
1
H(t, T0)
t∫
T0
H(t, s) (α1α2p(s) + r(s))
r2(s)v(s)
w2(s)ds ≤
≤ βα1
(β − 1)
(w(T0)− b(T0)) <∞. (2.28)
Now, we claim that
∞∫
T0
(α1α2p(s) + r(s))
r2(s)v(s)
w2(s)ds <∞. (2.29)
Suppose the contrary, that is,
∞∫
T0
(α1α2p(s) + r(s))
r2(s)v(s)
w2(s)ds =∞. (2.30)
It follows from (2.24) that there exists a positive constant δ such that
inf
s≥t0
[
lim inf
t→∞
H(t, s)
H(t, t0)
]
> δ. (2.31)
From (2.31),
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9
1274 E. TUNÇ, H. AVCI
lim inf
t→∞
H(t, s)
H(t, t0)
> δ > 0,
and there exists a T2 ≥ T1 such that H(t, T1)/H(t, t0) ≥ δ, for all t ≥ T2. On the other
hand, from (2.30), for any positive number κ, there exist a T1 > T0, such that, for all
t ≥ T1,
t∫
T0
(α1α2p(s) + r(s))
r2(s)v(s)
w2(s)ds ≥ κ
δ
.
Using integration by parts, we obtain that, for all t ≥ T1,
1
H(t, T0)
t∫
T0
H(t, s) (α1α2p(s) + r(s))
r2(s)v(s)
w2(s)ds =
=
1
H(t, T0)
t∫
T0
[
−∂H(t, s)
∂s
] s∫
T0
(α1α2p(τ) + r(τ))
r2(τ)v(τ)
w2(τ)dτ
ds ≥
≥ κ
δ
1
H(t, T0)
t∫
T1
[
−∂H(t, s)
∂s
]
ds =
κ
δ
H(t, T1)
H(t, T0)
. (2.32)
Hence, we have from (2.32)
1
H(t, T0)
t∫
T0
H(t, s) (α1α2p(s) + r(s))
r2(s)v(s)
w2(s)ds ≥ κ for all t ≥ T2.
Since κ is an arbitrary positive constant,
lim inf
t→∞
1
H(t, T0)
t∫
T0
H(t, s) (α1α2p(s) + r(s))
r2(s)v(s)
w2(s)ds = +∞,
which conradicts (2.28). Therefore, (2.29) holds, and from (2.27)
∞∫
T0
(α1α2p(s) + r(s))
r2(s)v(s)
b2+(s)ds ≤
∞∫
T0
(α1α2p(s) + r(s))
r2(s)v(s)
w2(s)ds < +∞,
which contradicts (2.26). Therefore, Eq. (1.1) is oscillatory.
Remark 1. In Theorem 5, the condition
lim sup
t→∞
1
H(t, t0)
t∫
t0
r2(s)v(s)
α1α2p(s) + r(s)
h2(t, s)ds <∞
is not necessary, however, an analogue of this condition is required in [21] (Theo-
rem 2.3).
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NEW OSCILLATION THEOREMS FOR A CLASS OF SECOND-ORDER DAMPED . . . 1275
Theorem 6. Let all assumptions of Theorem 5 satisfied except that condition (2.25)
be replaced with
lim inf
t→∞
1
H(t, T )
t∫
T
(
H(t, s)γ1(s)−
βα1r
2(s)v(s)
4 (α1α2p(s) + r(s))
h2(t, s)
)
ds ≥ b(T ).
Then Eq. (1.1) is oscillatory.
Proof. By an similar argument to that in the proof of Theorem 3, one can complete
the proof of this theorem. Therefore, we omit the detailed proof for the theorem.
Remark 2. In Theorem 6, the condition
lim inf
t→∞
1
H(t, t0)
t∫
t0
r2(s)v(s)
α1α2p(s) + r(s)
h2(t, s)ds <∞
is not necessary, however, an analogue of this condition is required in [21] (Theo-
rem 2.2).
3. Applications. Following the classical ideas of Kamenev [12], we define H(t, s)
as
H(t, s) = (t− s)n−1 , (t, s) ∈ D,
where n > 2 is an integer. Evidently, H ∈ P and
h(t, s) = (n− 1) (t− s)(n−3)/2 , (t, s) ∈ D.
Then, by Theorems 1 and 2 we have following two corollaries.
Corollary 3. Let (a) – (d) and (e1) hold. Suppose that there exists a function g ∈
∈ C1 ([t0,∞) ;R) such that, for some integer n > 2 and some β ≥ 1,
lim sup
t→∞
t1−n
t∫
t0
(t− s)n−3
(
(t− s)2 γ(s)− α1βr(s)v(s)
4
(n− 1)
2
)
ds =∞, (2.33)
where ν(t) and γ(t) are as in Theorem 1. Then Eq. (1.1) is oscillatory.
Corollary 4. Let assumptions (a) – (d) and (e1) be fulfilled. Assume that there exist
functions g ∈ C1 ([t0,∞) ;R) and b ∈ C ([t0,∞) ;R) such that, for all T ≥ t0, some
β > 1, and some integer n > 2,
lim sup
t→∞
t1−n
t∫
T
(
(t− s)n−1 γ(s)− α1β (n− 1)
2
4
r(s)v(s) (t− s)n−3
)
ds ≥ b(T ),
(2.34)
where γ(s), v(t) are as in Theorem 1. Suppose also that (2.12) is satisfied. Then Eq. (1.1)
is oscillatory.
Example 1. For t ≥ 1, consider the differential equation of the form(
x′
1 + x2
)
+
2(1 + cos t)
1 + x2
x′ +
(
2 + 2 cos t− sin t+ cos2 t
)
x
(
1 +
3
7 + x2
)
= 0.
(2.35)
It is easy to see that conditions (a) – (d) and (e1) hold with K = α1 = α2 = 1. Let
g(t) = 1 + cos t; then ν(t) = 1 and γ(t) = 1. Applying Corollary 3 with n = 3, we
have
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1276 E. TUNÇ, H. AVCI
lim sup
t→∞
1
t2
t∫
1
(
(t− s)2 · 1− β
)
ds =∞.
Hence, Eq. (2.35) is oscillatory by Corollary 3. Since condition (e) is not satisfied, the
results of [21] can not be used to Eq. (2.35).
Example 2. Consider the differential equation(
t2
x′
1 + x2
)
+
2t3
1 + x2
x′ +
(
6t2 − 6t2 sin2 t+ t4 + 2
)
x(2 + cosx) = 0, t ≥ 1,
(2.36)
where
k1(x, x
′) =
x′
1 + x2
, k2(x, x
′) =
1
1 + x2
(α1 = α2 = 1) and f(x) = x(2 + cosx) (K = 1).
We apply Corollary 4 with n = 3 and g(t) = t. Then, ν(t) = 1 and γ(t) = 3t2 −
− 6t2 sin2 t+ 2. Let β = 2. A direct computation yields
lim sup
t→∞
1
t2
t∫
T
(
(t− s)2
(
3s2 − 6s2 sin2 s+ 2
)
− 2s2
)
ds =
=
3
2
(
1 + cosT sinT − 2T cos2 T − T
3
− 2T 2 cosT sinT
)
= b(T ).
Let b+(t) = max (b(t), 0). The relation
b2+(t)
r(t)ν(t)
= O(t2) as t→∞
implies that the condition (2.12) is satisfied. Therefore, (2.36) is oscillatory by Corol-
lary 4. Note that in this example
lim sup
t→∞
1
t2
t∫
1
α1β
4
r(s)v(s)h2(t, s)ds = lim sup
t→∞
1
t2
t∫
1
2s2ds =∞. (2.37)
(2.37) show that we do not need to impose any condition similar to the condition
lim sup
t→∞
1
t2
t∫
1
α1β
4
r(s)v(s)h2(t, s)ds <∞
in Theorem 2, but the analogue of this condition is necessary for Theorem 3.4, 3.7
in [21].
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Received 10.01.11,
after revision — 18.07.11
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 9
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| id | umjimathkievua-article-2803 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| resource_txt_mv | umjimathkievua/ab/155263edb5697adef18fd7148a39a1ab.pdf |
| spelling | umjimathkievua-article-28032020-03-18T19:36:55Z Solvability of inhomogeneous boundary-value problems for fourth-order differential equations Новi осциляцiйнi теореми для одного класу згасаючих нелiнiйних диференцiальних рiвнянь другого порядку Avci, H. Tunç, E. Авкі, Х. Тунс, Е. Some new oscillation criteria are established for the nonlinear damped differential equation $$(r(t)k_1 (x, x'))' + p (t) k_2 (x, x') x' + q (t) f (x (t)) = 0,\quad t \geq t_0.$$ The results obtained extend and improve some existing results in the literature. Встановлено деякi новi осциляцiйнi критерiї для згасаючого нелiнiйного рiвняння $$(r(t)k_1 (x, x'))' + p (t) k_2 (x, x') x' + q (t) f (x (t)) = 0,\quad t \geq t_0.$$ Отриманi результати узагальнюють i посилюють деякi iснуючi результати. Institute of Mathematics, NAS of Ukraine 2011-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2803 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 9 (2011); 1263-1278 Український математичний журнал; Том 63 № 9 (2011); 1263-1278 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2803/2359 https://umj.imath.kiev.ua/index.php/umj/article/view/2803/2360 Copyright (c) 2011 Avci H.; Tunç E. |
| spellingShingle | Avci, H. Tunç, E. Авкі, Х. Тунс, Е. Solvability of inhomogeneous boundary-value problems for fourth-order differential equations |
| title | Solvability of inhomogeneous boundary-value problems for fourth-order differential equations |
| title_alt | Новi осциляцiйнi теореми для одного класу згасаючих нелiнiйних диференцiальних рiвнянь другого порядку |
| title_full | Solvability of inhomogeneous boundary-value problems for fourth-order differential equations |
| title_fullStr | Solvability of inhomogeneous boundary-value problems for fourth-order differential equations |
| title_full_unstemmed | Solvability of inhomogeneous boundary-value problems for fourth-order differential equations |
| title_short | Solvability of inhomogeneous boundary-value problems for fourth-order differential equations |
| title_sort | solvability of inhomogeneous boundary-value problems for fourth-order differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2803 |
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