On the Behavior of Solutions of a Third-Order Nonlinear Dynamic Equation on Time Scales
We study oscillatory and asymptotic properties of the third-order nonlinear dynamic equation $$ {{\left[ {{{{\left( {\frac{1}{{{r_2}(t)}}{{{\left( {{{{\left( {\frac{1}{{{r_1}(t)}}{x^{\varDelta }}(t)} \right)}}^{{{\gamma_1}}}}} \right)}}^{\varDelta }}} \right)}}^{{{\gamma_2}}}}} \right]}^{\varDelta...
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Institute of Mathematics, NAS of Ukraine
2013
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508381465280512 |
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| author | Şenel, M. T. Шенель, М. Т. |
| author_facet | Şenel, M. T. Шенель, М. Т. |
| author_sort | Şenel, M. T. |
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| collection | OJS |
| datestamp_date | 2020-03-18T19:16:28Z |
| description | We study oscillatory and asymptotic properties of the third-order nonlinear dynamic equation $$ {{\left[ {{{{\left( {\frac{1}{{{r_2}(t)}}{{{\left( {{{{\left( {\frac{1}{{{r_1}(t)}}{x^{\varDelta }}(t)} \right)}}^{{{\gamma_1}}}}} \right)}}^{\varDelta }}} \right)}}^{{{\gamma_2}}}}} \right]}^{\varDelta }}+f\left( {t,{x^{\sigma }}(t)} \right)=0,\quad t\in \mathbb{T}. $$ By using the Riccati transformation, we present new criteria for the oscillation or certain asymptotic behavior of solutions of this equation. It is supposed that the time scale T is unbounded above. |
| first_indexed | 2026-03-24T02:24:18Z |
| format | Article |
| fulltext |
UDC 517.9
M. T. Şenel (Erciyes Univ., Kayseri, Turkey)
ON THE BEHAVIOR OF SOLUTIONS
OF A THIRD-ORDER NONLINEAR DYNAMIC EQUATION ON TIME SCALES*
ПРО ПОВЕДIНКУ РОЗВ’ЯЗКIВ НЕЛIНIЙНОГО ДИНАМIЧНОГО РIВНЯННЯ
ТРЕТЬОГО ПОРЯДКУ НА ЧАСОВИХ ШКАЛАХ
The objective of this paper is to study oscillatory and asymptotic properties of the third-order nonlinear dynamic equation[(
1
r2(t)
((
1
r1(t)
x∆(t)
)γ1)∆)γ2]∆
+ f(t, xσ(t)) = 0, t ∈ T.
By using the Riccati transformation, we present new criteria for the oscillation or certain asymptotic behavior of solutions
of this equation. We suppose that the time scale T is unbounded above.
Метою цiєї статтi є вивчення осциляцiйних та асимптотичних властивостей нелiнiйного динамiчного рiвняння
третього порядку [(
1
r2(t)
((
1
r1(t)
x∆(t)
)γ1)∆)γ2]∆
+ f(t, xσ(t)) = 0, t ∈ T.
За допомогою перетворення Рiккатi отримано новi критерiї осциляцiї та певної асимптотичної поведiнки розв’язкiв
цього рiвняння. Часова шкала T вважається необмеженою зверху.
1. Preliminaries and notation. Much recent attention has been given to dynamic equations on
time scales, or measure chains, and we refer the reader to the landmark paper of S. Hilger [1] for
a comprehensive treatment of the subject. Since then, several authors have expounded on various
aspects of this new theory; see the survey paper by Agarwal, Bohner, O’Regan and Peterson [2].
A book on the subject of time scales by Bohner and Peterson [3] also summarizes and organizes
much of the time scale calculus. The various type oscillation and nonoscillation criteria for solutions
of ordinary and dynamic equations have been studied extensively in a large cycle of works (see
[4 – 12]).
In [4], the authors have considered third-order nonlinear dynamic equation (1.1) for γ1 = γ2 = 1.
They have studied asymptotic behavior that equation. Yu and Wang [5] have considered the third-
order nonlinear dynamic equation
(1/(a2(t)(((1/(a1(t))(x∆(t))α1)∆)α2)∆ + q(t)f(x(t)) = 0,
where α1 and α2 are quotient of odd positive integers. They have supposed that a1, a2 and q are
positive, real-valued, rd-continuous functions defined on time scale T.
f ∈ C(R,R) is assumed to satisfy xf(x) > 0 (x 6= 0), and for k > 0, ∃M = Mk > 0,
f(x)
x
≥M, |x| ≥ k. The authors have studied the asymptotic behavior of solution of above equation.
A time scale T is an arbitrary nonempty closed subset of the real numbers R. The forward and
the backward jump operators on any time scale T are defined by σ(t) := inf{s ∈ T : s > t},
ρ(t) := sup{s ∈ T : s < t}. A point t ∈ T, t > inf T, is said to be left-dense if ρ(t) = t, right-dense
if t < supT and σ(t) = t, left-scattered if ρ(t) < t and right-scattered if σ(t) > t. The graininess
* This work was supported by the Research Fund of the Erciyes University (Project No. FBA-11-3391).
c© M. T. ŞENEL, 2013
996 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
ON THE BEHAVIOR OF SOLUTIONS OF A THIRD-ORDER NONLINEAR DYNAMIC EQUATION . . . 997
function µ for a time scale T is defined by µ(t) := σ(t) − t. For a function f : T → R the (delta)
derivative is defined by
f∆(t) =
f(σ(t))− f(t)
σ(t)− t
,
if f is continuous at t and t is right-scattered. If t is not right-scattered, then the derivative is defined
by
f∆(t) = lim
s→t+
f(σ(t))− f(s)
σ(t)− s
= lim
s→t+
f(t)− f(s)
t− s
,
provided this limit exists. A function f : [a, b] → R is said to be right-dense continuous if it is right
continuous at each right-dense point and there exists a finite left limit at all left-dense points, and f
is said to be differentiable if its derivative exists. A useful formula dealing with the time scale is that
fσ = f(σ(t)) = f(t) + µ(t)f∆(t).
We will make use of the following product and quotient rules for the derivative of the product fg
and the quotient f/g (where ggσ 6= 0) of two differentiable functions f and g
(fg)∆ = f∆g + fσg∆ = fg∆ + f∆gσ,
(
f
g
)∆
=
f∆g − fg∆
ggσ
.
The integration by parts formula is
b∫
a
f∆(t)g(t)∆t = f(b)g(b)− f(a)g(a)−
b∫
a
fσ(t)g∆(t)∆(t).
The function f : T→ R is called rd-continuous if it is continuous at the right-dense points and if the
left-sided limits exist in left-dense points. Not only does the new theory of the so-called “dynamic
equations” unify the theories of differential and difference equations, but also extends these classical
cases to cases “in between”, e.g., to the so-called q-difference equations when T = qN = {qt : t ∈ N0
for q > 1} (which has important application in quantum theory).
We will study the asymptotic behavior or oscillation of solutions of third-order nonlinear dynamic
equation [(
1
r2(t)
((
1
r1(t)
x∆(t)
)γ1
)∆
)γ2
]∆
+ f(t, xσ(t)) = 0 t ∈ T, (1.1)
or for short,
L3x(t) + f(t, xσ(t)) = 0, t ∈ T, (1.2)
where T is a time scale,
L1x(t) =
(
1
r1(t)
x∆(t)
)γ1
,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
998 M. T. ŞENEL
L2x(t) =
(
1
r2(t)
((
1
r1(t)
x∆(t)
)γ1
)∆
)γ2
,
L3x(t) = [L2x(t)]∆.
In the sequel we will assume:
(H1) rn(t) are positive, real-value, rd-continuous functions defined on the time scales such that
∞∫
T0
rn(s)∆s =∞, n = 1, 2;
(H2) γn is a quotient of odd positive integers, n = 1, 2;
(H3) f : T × R → R is continuous function, there exists k > 0 such that uf(t, u) > 0, u 6= 0,
f(t, u)
u
≥ q(t), |u| ≥ k. q(t) is positive, real-valued, rd-continuous function defined on time scales.
A solution x(t) of equation (1.1) is said to be oscillatory if it is neither eventually positive nor
eventually negative, otherwise it is nonoscillatory.
2. Main results. We need the following lemmas which play an important role in the proof of
main results.
Lemma 2.1. Assume (H1) – (H3) hold, x(t) is an eventually positive solution of (1.1). Then
there exists a T1 ∈ [T0,∞) such either:
(i) x(t) > 0, L1x(t) > 0, L2x(t) > 0, t ∈ [T1,∞),
or
(ii) x(t) > 0, L1x(t) < 0, L2x(t) > 0, t ∈ [T1,∞).
Proof. Let x(t) be a eventually positive solution of (1.1), then there exists T1 ∈ [T0,∞) such
that x(t) > 0 for t ∈ [T1,∞). Since xσ(t) > x(t) > 0, f(t, xσ(t)) > 0 for t ∈ [T1,∞) and from
(1.1) we have
L3x(t) = −f(t, xσ(t)) < 0, t ∈ [T1,∞),
which implies that L2x(t) is strictly decreasing on t ∈ [T1,∞).We claim that L2x(t) > 0. Otherwise,
there exists a T2 ∈ [T1,∞) such that
L2x(t) ≤ L2x(T2) < 0, t ∈ [T2,∞),
that is, [
1
r2(t)
((
1
r1(t)
x∆(t)
)γ1
)∆
]γ2
≤ L2x(T2) < 0, t ∈ [T2,∞).
Hence we have
(L1x(t))∆ ≤ r2(t)(L2x(T2))
1
γ2 (2.1)
which implies that L1x(t) is strictly decreasing on [T2,∞). Integrating (2.1) from T2 to t, we obtain
L1x(t) ≤ L1x(T2) + (L2x(T2))
1
γ2
t∫
T2
r2(s)∆s.
Letting t→∞, we have L1x(t)→ −∞. Thus, there exists T3 ∈ [T2,∞) such that
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
ON THE BEHAVIOR OF SOLUTIONS OF A THIRD-ORDER NONLINEAR DYNAMIC EQUATION . . . 999
L1x(t) ≤ L1x(T3) < 0, t ∈ [T3,∞),
that is, (
1
r1(t)
x∆(t)
)γ1
≤ L1x(T3) < 0, t ∈ [T3,∞).
It follows that
x∆(t) ≤ (L1x(T3))
1
γ1 r1(t), t ∈ [T3,∞).
Integrating from T3 to t, we have
x(t) ≤ x(T3) + (L1x(T3))
1
γ1
t∫
T3
r1(s)∆s.
Letting t → ∞, we have x(t) → −∞, which is a contradiction with the fact that x(t) > 0. Hence
L2x(t) > 0, t ∈ [T1,∞). This implies that L1x(t) is strictly increasing on [T1,∞). It follows that
either L1x(t) > 0 or L1x(t) < 0.
Lemma 2.1 is proved.
Lemma 2.2. Assume (H1) – (H3) and
∫ ∞
T0
q(s)∆s = ∞ hold. If x(t) is a solution of (1.1)
that satisfies Case (ii) in Lemma 2.1, then limt→∞ x(t) = 0.
Proof. Suppose that x(t) be solution of (1.1) satisfying case (ii) in Lemma 2.1. Then from
L1x(t) < 0, we get (
1
r1(t)
x∆(t)
)γ1
< 0, t ≥ T1.
So, x∆(t) < 0 for t ≥ T1 and limt→∞ x(t) = b ≥ 0. We claim that b = 0. Assume not, that is, let be
x(t) ≥ b > 0, t ≥ T1. With k = b, from (H3) and xσ(t) > x(t),
L3x(t) = −f(t, xσ(t)) ≤ −q(t)xσ(t) ≤ −q(t)x(t) < −bq(t), t ≥ T1.
Letting y(t) := L2x(t) > 0, t ≥ T1, then
y∆(t) = L3x(t) < −q(t)b, t ≥ T1.
Integrating the last inequality from T1 to t, we have
y(t) ≤ y(T1)− b
t∫
T1
q(s)∆s.
Letting t → ∞, we have y(t) → −∞, which is a contradiction. Therefore, b = 0, that is,
limt→∞ x(t) = 0.
Lemma 2.2 is proved.
Lemma 2.3. Suppose that (H1) – (H3) hold. If x(t) is a solution of (1.1) satisfying Case (i)
of Lemma 2.1, then there exists T1 ∈ [T0,∞) such that
L1x(t) ≥ R(t, T1)(L2x(t))
1
γ2
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
1000 M. T. ŞENEL
or
x∆(t) ≥ r1(t)(R(t, T1))
1
γ1 (L2x(t))
1
γ1γ2 ,
and
L1x(t)
R(t, T1)
is decreasing on (T1,∞), where R(t, T1) =
∫ t
T1
r2(s)∆s.
Proof. Let x(t) be a solution of (1.1) satisfying case (i) of Lemma 2.1. Then from (1.1) we
have L3x(t) < 0 for t ∈ [T1,∞), so L2x(t) strictly decreasing on [T1,∞). From L2x(t) =
=
(
1
r2(t)
((
1
r1(t)
x∆(t)
)γ1
)∆
)γ2
, we obtain
(L1x(t))∆ = r2(t)(L2x(t))
1
γ2 .
Then for t ≥ T1, we have
t∫
T1
(L1x(s))∆∆s = L1x(t)− L1x(T1) =
=
t∫
T1
r2(s)(L2x(s))
1
γ2 ∆s ≥ (L2x(t))
1
γ2
t∫
T1
r2(s)∆s.
It follows that (
1
r1(t)
x∆(t)
)γ1
= L1x(t) ≥ L1x(T1) +R(t, T1)(L2x(t))
1
γ2 ≥
≥ R(t, T1)(L2x(t))
1
γ2 , t ≥ T1, (2.2)
so, we get
x∆(t) ≥ r1(t)(R(t, T1))
1
γ1 ((L2x(t))
1
γ1γ2 , t ≥ T1.
We claim that
L1x(t)
R(t, T1)
is decreasing on (T1,∞). For t > T1, from (2.2) we get
(
L1x(t)
R(t, T1)
)∆
=
(L1x(t))∆R(t, T1)− L1x(t)(R(t, T1))∆
R(t, T1)R(σ(t), T1)
=
=
r2(t)(L2x(t))
1
γ2R(t, T1)− L1x(t)r2(t)
R(t, T1)R(σ(t), T1)
≤ r2(t)(L1x(t)− L1x(t))
R(t, T1)R(σ(t), T1)
= 0.
Hence,
L1x(t)
R(t, T1)
is decreasing on (T1,∞).
Lemma 2.3 is proved.
Theorem 2.1. Suppose that (H1) – (H3) and γ1γ2 = 1 hold, and assume that there exists a
positive function z(t) such that z∆ is rd-continuous on [T0,∞), we have
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
ON THE BEHAVIOR OF SOLUTIONS OF A THIRD-ORDER NONLINEAR DYNAMIC EQUATION . . . 1001
lim sup
t→∞
t∫
T0
[
z(s)q(s)− (z∆(s))2
4B(s, T1)
]
∆s =∞, t > T1 ≥ T0, (2.3)
where B(t, T1) = z(t)r1(t)(R(t, T1))
1
γ1 , R(t, T1) =
∫ t
T1
r2(s)∆s. Then every solution x(t) of (1.1)
is either oscillatory or limt→∞ x(t) exists.
Proof. Let x(t) be a nonoscillatory solution of (1.1). Assume that x(t) is eventually positive
(the case when x(t) < 0 is similar). By Lemma 2.1 we see that x(t) satisfies either case (i) or (ii).
We claim that case (i) of Lemma 2.1 is not true. Assume not, there exists T1 ∈ [T0,∞), such that
x(t) > 0, L1x(t) > 0, L2x(t) > 0 for t > T1. Consider the Riccati substitution
w(t) = z(t)
L2x(t)
x(t)
> 0, t ∈ [T1,∞). (2.4)
From (1.1) we obtain
w∆(t) =
[
z(t)
x(t)
]∆
(L2x(t))σ +
z(t)
x(t)
(L2x(t))∆ =
=
z∆(t)x(t)− z(t)x∆(t)
x(t)xσ(t)
(L2x(t))σ − z(t)
x(t)
f(t, xσ(t)).
Since, x∆(t) > 0, xσ(t) > x(t), and L2x(t) ≥ (L2x(t))σ, t ≥ T1 and Lemma 2.3, we get
w∆(t) ≤ z∆(t)
(L2x(t))σ
xσ(t)
− z(t) x∆(t)
(xσ(t))2
(L2x(t))σ − z(t)f(t, xσ(t))
xσ(t)
≤
≤ −z(t)q(t) + z∆(t)
wσ(t)
zσ(t)
− z(t)r1(t)(R(t, T1))
1
γ1L2x(t)(L2x(t))σ
(xσ(t))2
=
= −z(t)q(t) + z∆(t)
wσ(t)
zσ(t)
−B(t, T1)
(wσ(t))2
(zσ(t))2
=
= −z(t)q(t) +
(z∆(t))2
4B(t, T1)
−
[√
B(t, T1)
wσ(t)
zσ(t)
− z∆(t)
2
√
B(t, T1)
]2
≤
≤ −z(t)q(t) +
(z∆(t))2
4B(t, T1)
,
hence we have
w∆(t) ≤ −
[
z(t)q(t)− (z∆(t))2
4B(t, T1)
]
. (2.5)
Integrating (2.5) from T2 to t, we find that
w(t)− w(T2) ≤ −
t∫
T2
[
z(s)q(s)− (z∆(s))2
4B(s, .)
]
∆s ≥ −w(T2),
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
1002 M. T. ŞENEL
that is
t∫
T2
[
z(s)q(s)− (z∆(s))2
4B(s, .)
]
∆s ≤ w(T2),
which is contradiction with (2.3). Hence, case (i) of Lemma 2.1 is not true. If case (ii) of Lemma 2.1
holds, then clearly limt→∞ x(t) exists.
Theorem 2.1 is proved.
Corollary 2.1. Suppose that (H1) – (H3) and γ1γ2 = 1 hold. If
∞∫
T0
q(s)∆s =∞, (2.6)
then every solution x(t) of (1.1) is either oscillatory or limt→∞ x(t) = 0.
Proof. If we take z(t) = 1 in Theorem 2.1, by the proof of Theorem 2.1 we have that every
solution x(t) of (1.1) is either oscillatory or limt→∞ x(t) exists. For the last case, by Lemma 2.2 we
obtain limt→∞ x(t) = 0.
Corollary 2.2. Suppose that (H1) – (H3) and γ1γ2 = 1 hold. If
lim sup
t→∞
t∫
T0
[
sq(s)− 1
4sr1(s)(R(s, T1))
1
γ1
]
∆s =∞, (2.7)
then every solution x(t) of (1.1) is either oscillatory or limt→∞ x(t) exists.
Proof. If we take z(t) = t in Theorem 2.1, by the proof of Theorem 2.1 we have that every
solution x(t) of (1.1) is either oscillatory or limt→∞ x(t) exists.
Corollary 2.2 is proved.
Example 2.1. Consider the equation((1
t
x∆(t)
) 1
3
)∆
3∆
+
1
t
|xσ(t)| = 0, (2.8)
where t ∈ T = qN0 , q0 > 1, r2(t) = 1, r1(t) = 1/t and f(t, xσ(t)) = 1/t|xσ(t)|, q(t) ≥ 1
t
, γ1 = 1/3,
γ2 = 3. For sufficient large T1,
R(t, T1) =
t∫
T1
∆s = t− T1
and for T2 > T1,
lim sup
t→∞
t∫
T2
[
sq(s)− 1
4sr1(s)(R(s, T1))
1
γ1
]
∆s = lim sup
t→∞
t∫
T2
[
s
1
s
− 1
4s1
s (s− T1)3
]
∆s =∞.
We get that all conditions of Corollary 2.2 are satisfied and then every solution x(t) of (2.8) is either
oscillatory or limt→∞ x(t) exists.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
ON THE BEHAVIOR OF SOLUTIONS OF A THIRD-ORDER NONLINEAR DYNAMIC EQUATION . . . 1003
Corollary 2.3. Suppose that (H1) – (H3) and γ1γ2 = 1 hold. If there is α > 1 such that
lim sup
t→∞
t∫
T0
[
sαq(s)− ((sα)∆)2
4sαr1(s)(R(s, T1))
1
γ1
]
∆s =∞, (2.9)
then every solution x(t) of (1.1) is either oscillatory or limt→∞ x(t) exists.
Proof. We take z(t) = tα, α > 1 in Theorem 2.1, by the proof of Theorem 2.1 we have that
every solution x(t) of (1.1) is either oscillatory or limt→∞ x(t) exists.
Corollary 2.3 is proved.
Theorem 2.2. Assume (H1) – (H3) and γ1γ2 = 1 hold. If there exist m ≥ 1 and a positive
function z(t) such that z∆ is rd-continuous on [T0,∞) such that
lim sup
t→∞
1
tm
t∫
T0
(t− s)m
[
z(s)q(s)− (z∆(s))2
4B(t, T1)
]
∆s =∞, (2.10)
where B(t, T1) = z(t)r1(t)(R(t, T1))
1
γ1 , R(t, T1) =
∫ t
T1
r2(s)∆s. Then every solution x(t) of (1.1)
is either oscillatory or limt→∞ x(t) exists.
Proof. Proceeding as in Theorem 2.1, we suppose that (1.1) has a nonoscillatory solution. Let
be x(t) > 0, t ≥ T1. Multiplying (2.5) by (t− s)m (with t replaced by s) and then integrating from
T2 to t (t ≥ T2 > T1), we have
t∫
T2
(t− s)mw∆(s)∆s ≤ −
t∫
T2
(t− s)m
[
z(s)q(s)− (z∆(s))2
4B(t, T1)
]
∆s.
An integrating by parts of left-hand side leads to
t∫
T2
(t− s)mw∆(s)∆s = (t− s)mw(s)
∣∣∣∣t
T2
−
t∫
T2
((t− s)m)∆sw(σ(s))∆s.
Let be h(t, s) := ((t− s)m)∆s . Since
h(t, s) =
−m(t− s)m−1, µ(s) = 0,
(t− σ(s))m − (t− s)m
µ(s)
, µ(s) > 0,
and when m ≥ 1 for t ≥ σ(s), it follows that
t∫
T2
(t− s)mw∆(s) ≥ −(t− T2)mw(T2)
or
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
1004 M. T. ŞENEL
1
tm
t∫
T2
(t− s)m
[
z(s)q(s)− (z∆(s))2
4B(t, T1)
]
∆s ≤
(
t− T2
t
)m
w(T2) ≤ w(T2),
a contradiction with (2.10). Thus, case (i) in Lemma 2.1 is not true. If case (ii) in Lemma 2.1 holds,
then as before, limt→∞ x(t) exists.
Theorem 2.2 is proved.
Corollary 2.4. Suppose that (H1) – (H2) and γ1γ2 = 1 hold. If there exist m ≥ 1,
lim sup
t→∞
1
tm
t∫
T0
(t− s)mq(s)∆s =∞,
then every solution x(t) of (1.1) is either oscillatory or limt→∞ x(t) = 0.
Proof. If we take z(t) = 1 in Theorem 2.2, by the proof of Theorem 2.2 we have that every
solution x(t) of (1.1) is either oscillatory or limt→∞ x(t) exists. For the last case, by Lemma 2.2 we
obtain limt→∞ x(t) = 0.
Corollary 2.4 is proved.
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Received 05.04.12,
after revision — 16.02.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
|
| id | umjimathkievua-article-2484 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:24:18Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c6/4e77a34c53a27d492bb392f2a0bdecc6.pdf |
| spelling | umjimathkievua-article-24842020-03-18T19:16:28Z On the Behavior of Solutions of a Third-Order Nonlinear Dynamic Equation on Time Scales Про поведшку розв'язків нелінійного динамічного рівняння третього порядку на часових шкалах Şenel, M. T. Шенель, М. Т. We study oscillatory and asymptotic properties of the third-order nonlinear dynamic equation $$ {{\left[ {{{{\left( {\frac{1}{{{r_2}(t)}}{{{\left( {{{{\left( {\frac{1}{{{r_1}(t)}}{x^{\varDelta }}(t)} \right)}}^{{{\gamma_1}}}}} \right)}}^{\varDelta }}} \right)}}^{{{\gamma_2}}}}} \right]}^{\varDelta }}+f\left( {t,{x^{\sigma }}(t)} \right)=0,\quad t\in \mathbb{T}. $$ By using the Riccati transformation, we present new criteria for the oscillation or certain asymptotic behavior of solutions of this equation. It is supposed that the time scale T is unbounded above. Метою цієї статті є вивчення осциляційних та асимптотичних властивостей нєлінійного динамiчного рівняння третього порядку $$ {{\left[ {{{{\left( {\frac{1}{{{r_2}(t)}}{{{\left( {{{{\left( {\frac{1}{{{r_1}(t)}}{x^{\varDelta }}(t)} \right)}}^{{{\gamma_1}}}}} \right)}}^{\varDelta }}} \right)}}^{{{\gamma_2}}}}} \right]}^{\varDelta }}+f\left( {t,{x^{\sigma }}(t)} \right)=0,\quad t\in \mathbb{T}. $$ За допомогою перетворення Ріккаті отримано нові критерії осциляції та певної асимптотичної поведінки розв'язків цього рівняння. Часова шкала T вважається необмеженою зверху. Institute of Mathematics, NAS of Ukraine 2013-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2484 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 7 (2013); 996–1004 Український математичний журнал; Том 65 № 7 (2013); 996–1004 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2484/1734 https://umj.imath.kiev.ua/index.php/umj/article/view/2484/1735 Copyright (c) 2013 Şenel M. T. |
| spellingShingle | Şenel, M. T. Шенель, М. Т. On the Behavior of Solutions of a Third-Order Nonlinear Dynamic Equation on Time Scales |
| title | On the Behavior of Solutions of a Third-Order Nonlinear Dynamic Equation on Time Scales |
| title_alt | Про поведшку розв'язків нелінійного динамічного рівняння третього порядку на часових шкалах |
| title_full | On the Behavior of Solutions of a Third-Order Nonlinear Dynamic Equation on Time Scales |
| title_fullStr | On the Behavior of Solutions of a Third-Order Nonlinear Dynamic Equation on Time Scales |
| title_full_unstemmed | On the Behavior of Solutions of a Third-Order Nonlinear Dynamic Equation on Time Scales |
| title_short | On the Behavior of Solutions of a Third-Order Nonlinear Dynamic Equation on Time Scales |
| title_sort | on the behavior of solutions of a third-order nonlinear dynamic equation on time scales |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2484 |
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