Note on the oscillation of second-order quasilinear neutral dynamic equations on time scales
By using a new method, we improve some results from [Saker S. H. Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales // Nonlin. Oscillations. – 2011. – 13, № 3. – P. 407 – 428]. З використанням нового методу покращено деякi результати, одержанi в ро...
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| Date: | 2018 |
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Інститут математики НАН України
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| Cite this: | Note on the oscillation of second-order quasilinear neutral dynamic equations on time scales / Chenghui Zhang, Tongxing Li // Нелінійні коливання. — 2018. — Т. 21, № 4. — С. 470-472 — Бібліогр.: 1 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860250369063387136 |
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| author | Chenghui Zhang Tongxing Li |
| author_facet | Chenghui Zhang Tongxing Li |
| citation_txt | Note on the oscillation of second-order quasilinear neutral dynamic equations on time scales / Chenghui Zhang, Tongxing Li // Нелінійні коливання. — 2018. — Т. 21, № 4. — С. 470-472 — Бібліогр.: 1 назв. — англ. |
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| description | By using a new method, we improve some results from [Saker S. H. Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales // Nonlin. Oscillations. – 2011. – 13, № 3. – P. 407 – 428].
З використанням нового методу покращено деякi результати, одержанi в роботi [Saker S. H. Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales // Нелiн. коливання. – 2010. – 13, № 3. – С. 379–399].
|
| first_indexed | 2025-12-07T18:42:15Z |
| format | Article |
| fulltext |
UDC 517.956.3
NOTE ON THE OSCILLATION OF SECOND-ORDER QUASILINEAR NEUTRAL
DYNAMIC EQUATIONS ON TIME SCALES
ПРО КОЛИВАННЯ КВАЗIЛIНIЙНИХ НЕЙТРАЛЬНИХ ДИНАМIЧНИХ РIВНЯНЬ
ДРУГОГО ПОРЯДКУ НА ЧАСОВИХ МАСШТАБАХ
Ch. Zhang
Shandong Univ., School Control Sci. and Eng.
Jinan, Shandong, 250061, P. R. China
e-mail: zchui@sdu.edu.cn
T. Li
Shandong Univ., School Control Sci. and Eng.
Jinan, Shandong, 250061, P. R. China
and Univ. Jinan, School Math. Sci.
Jinan, Shandong, 250022, P. R. China
e-mail: litongx2007@163.com
By using a new method, we improve some results from [Saker S. H. Oscillation criteria for a second-order
quasilinear neutral functional dynamic equation on time scales // Nonlin. Oscillations. – 2011. – 13, № 3. –
P. 407 – 428].
З використанням нового методу покращено деякi результати, одержанi в роботi [Saker S. H. Osci-
llation criteria for a second-order quasilinear neutral functional dynamic equation on time scales // Нелiн.
коливання. – 2010. – 13, № 3. – С. 379–399].
1. Introduction. In 2011, Saker [1] established some sufficient conditions for the oscillation of
the second-order quasilinear neutral functional dynamic equation(
p(t)
(
(y(t) + r(t)y(τ(t)))∆
)γ)∆
+ f(t, y(δ(t))) = 0, t ∈ [t0,∞)T, (1.1)
for which is assumed the following hypotheses:
(h1 ) γ > 0 is the quotient of odd positive integers, r and p are real-valued rd-continuous
positive functions defined on T, τ, δ : [t0,∞)T → T, τ(t) ≤ t, and limt→∞ τ(t) = limt→∞ δ(t) =
=∞;
(h2 ) 0 ≤ r(t) < 1;
(h3 ) f(t, u) : T × R → R is a continuous function such that uf(t, u) > 0 for all u 6= 0 and
there exists a positive rd-continuous function q(t) defined on T such that |f(t, u)| ≥ q(t)|uβ|,
where β > 0 is a ratio of odd positive integers.
Under the condition
∞∫
t0
1
p
1
γ (t)
∆t <∞ (1.2)
© Ch. Zhang, T. Li, 2018
470 ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 4
NOTE ON THE OSCILLATION OF SECOND-ORDER QUASILINEAR NEUTRAL . . . 471
and the assumptions
δ(t) ≤ τ(t) ≤ t, τ∆(t) ≥ 0, r∆(t) ≥ 0, (1.3)
Saker [1] obtained some new oscillation criteria for (1.1); see [1] (Section 3). In the last section
of the paper [1], the author posed a problem: How to present oscillation criteria for (1.1) when
condition (1.3) does not hold?
By a solution of (1.1), we mean a nontrivial real-valued function y satisfying (1.1) for
t ∈ T. We recall that a solution y of (1.1) is said to be oscillatory on [t0,∞)T if it is neither
eventually positive nor eventually negative; otherwise, the solution is said to be nonoscillatory.
Equation (1.1) is said to be oscillatory if all its solutions are oscillatory. Our attention is restricted
to those solutions y of (1.1) which are not eventually identically zero.
Our aim in this paper is to give an answer for the problem posed by [1].
In what follows, all functional inequalities considered in this note are assumed to hold
eventually, that is, they are satisfied for all t large enough.
2. Main results. Note that [1] (Eq. (3.7)) plays an important role in the obtained results
of [1] (Section 3). Hence, we will change it in order to renew results of [1]. Now we give the
following. We let
x(t) := y(t) + r(t)y(τ(t)), P (t) :=
∞∫
t
1
p
1
γ (s)
∆s, 1− r(t) P (τ(t))
P (t)
> 0.
Lemma 1. Let (1.2) hold, δ(t) ≤ t, and y be an eventually positive solution of (1.1). Assume
further that
(
p
(
x∆
)γ)∆
(t) < 0, x∆(t) < 0, x(t) > 0 for t ∈ [t0,∞)T. Then
(
p
(
x∆
)γ)∆
(t) + q(t)
(
1− r(δ(t)) P (τ(δ(t)))
P (δ(t))
)β
xβ(t) ≤ 0. (2.1)
Proof. From
(
p
(
x∆
)γ)∆
(t) < 0, we have
x∆(s) ≤ p
1
γ (t)
p
1
γ (s)
x∆(t), s ≥ t.
Integrating this from t to `, we obtain
x(`) ≤ x(t) + p
1
γ (t)x∆(t)
`∫
t
1
p
1
γ (s)
∆s.
Letting `→∞, we have
x(t) ≥ −P (t)p
1
γ (t)x∆(t).
Hence
( x
P
)∆
(t) =
x∆(t)P (t)− x(t)P∆(t)
P (t)P (σ(t))
=
x∆(t)P (t) +
x(t)
p
1
γ (t)
P (t)P (σ(t))
≥ 0,
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 4
472 CH. ZHANG, T. LI
which yields
y(t) = x(t)− r(t)y(τ(t)) ≥ x(t)− r(t)x(τ(t)) ≥
(
1− r(t) P (τ(t))
P (t)
)
x(t).
Thus, from (1.1), we have(
p
(
x∆
)γ)∆
(t) + q(t)
(
1− r(δ(t)) P (τ(δ(t)))
P (δ(t))
)β
xβ(δ(t)) ≤ 0,
which follows from x∆(t) < 0 and δ(t) ≤ t that (2.1) holds. The proof is complete.
Following ideas of [1] (Theorem 3.1) and Lemma 1 in this note, we can renew [1] (Eq. (3.7))
by the following:
∞∫
T
1
p(s)
s∫
T
g∗(u)P β(u)∆u
1
γ
∆s =∞, (2.2)
where
g∗(u) := q(u)
(
1− r(δ(u))
P (τ(δ(u)))
P (δ(u))
)β
.
Therefore, replacing [1] (Eq. (3.7)) and (1.3) with (2.2) and δ(t) ≤ t in this paper, we can
renew [1] (Theorem 3.1, Theorem 3.2, Theorem 3.3, Theorem 3.4, Theorem 3.5). The details are
left to the reader.
3. Acknowledgements. The authors would like to thank the referees for giving useful
suggestions and comments for the improvement of this paper. This research is supported by
NNSF of PR China (Grant Nos. 61034007, 60874016, 50977054). The second author would
like to express his gratitude to Professors Ravi P. Agarwal and Martin Bohner for their selfless
guidance.
References
1. Saker S. H. Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time
scales // Nonlin. Oscillations. – 2011. – 13, № 3. – P. 407 – 428.
Received 20.10.11,
after revision — 22.10.11
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т. 21, № 4
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| id | nasplib_isofts_kiev_ua-123456789-177340 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-12-07T18:42:15Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chenghui Zhang Tongxing Li 2021-02-14T11:33:34Z 2021-02-14T11:33:34Z 2018 Note on the oscillation of second-order quasilinear neutral dynamic equations on time scales / Chenghui Zhang, Tongxing Li // Нелінійні коливання. — 2018. — Т. 21, № 4. — С. 470-472 — Бібліогр.: 1 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/177340 517.956.3 By using a new method, we improve some results from [Saker S. H. Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales // Nonlin. Oscillations. – 2011. – 13, № 3. – P. 407 – 428]. З використанням нового методу покращено деякi результати, одержанi в роботi [Saker S. H. Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales // Нелiн. коливання. – 2010. – 13, № 3. – С. 379–399]. en Інститут математики НАН України Нелінійні коливання Note on the oscillation of second-order quasilinear neutral dynamic equations on time scales Про коливання квазілінійних нейтральних динамічних рівнянь другого порядку на часових шкалах О колебаниях квазилинейных нейтральных динамических уравнений второго порядка на временных шкалах Article published earlier |
| spellingShingle | Note on the oscillation of second-order quasilinear neutral dynamic equations on time scales Chenghui Zhang Tongxing Li |
| title | Note on the oscillation of second-order quasilinear neutral dynamic equations on time scales |
| title_alt | Про коливання квазілінійних нейтральних динамічних рівнянь другого порядку на часових шкалах О колебаниях квазилинейных нейтральных динамических уравнений второго порядка на временных шкалах |
| title_full | Note on the oscillation of second-order quasilinear neutral dynamic equations on time scales |
| title_fullStr | Note on the oscillation of second-order quasilinear neutral dynamic equations on time scales |
| title_full_unstemmed | Note on the oscillation of second-order quasilinear neutral dynamic equations on time scales |
| title_short | Note on the oscillation of second-order quasilinear neutral dynamic equations on time scales |
| title_sort | note on the oscillation of second-order quasilinear neutral dynamic equations on time scales |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/177340 |
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