On Agarwal - Pang-type integral inequalities
We establish some new Agarwal – Pang-type inequalities involving second-order partial derivatives. Our results in special cases yield some of interrelated results and provide new estimates for inequalities of this type.
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| author | Bencze, M. Zhao, C. J Бенче, М. Чжао, С. Дж. |
| author_facet | Bencze, M. Zhao, C. J Бенче, М. Чжао, С. Дж. |
| author_sort | Bencze, M. |
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| description | We establish some new Agarwal – Pang-type inequalities involving second-order partial derivatives. Our results in special
cases yield some of interrelated results and provide new estimates for inequalities of this type. |
| first_indexed | 2026-03-24T02:25:56Z |
| format | Article |
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UDC 517.5
C. J. Zhao (China Jiliang Univ., Hangzhou, China),
M. Bencze (Braşov, Romania)
ON AGARWAL – PANG-TYPE INTEGRAL INEQUALITIES*
ПРО НЕРIВНОСТI ТИПУ АГАРВАЛА – ПАНГА
We establish some new Agarwal – Pang-type inequalities involving second-order partial derivatives. Our results in special
cases yield some of interrelated results and provide new estimates for inequalities of this type.
Встановлено деякi новi нерiвностi типу Агарвала – Панга, що мiстять частиннi похiднi другого порядку. В окремих
випадках iз одержаних результатiв випливають деякi пов’язанi результати та новi оцiнки для нерiвностей цього
типу.
1. Introduction. In the year 1960, Opial [1] established the following inequality:
Theorem A. Suppose f ∈ C1[0, h] satisfies f(0) = f(h) = 0 and f(x) > 0 for all x ∈ (0, h).
Then the inequality holds
h∫
0
∣∣f(x)f ′(x)
∣∣ dx ≤ h
4
h∫
0
(f ′(x))2dx, (1.1)
where this constant h/4 is best possible.
Many generalizations, extensions and discretizations of Opial’s inequality were established (see
e.g. [2 – 15]). For an extensive survey on these inequalities, see [16]. Opial’s inequality and its
generalizations and extensions play a fundamental role in establishing the existence and uniqueness
of initial and boundary-value problems for ordinary and partial differential equations as well as
difference equation [8 – 16].
In 1995, Agarwal and Pang [17] proved the following Wirtinger’s type inequality and an inter-
esting Opial’s type inequality, respectively.
Theorem B. Let λ ≥ 1 be a given real number, and let p(t) be a nonnegative and continuous
function on [0, a]. Further, let x(t) be an absolutely continuous function on [0, a], with x(0) = x(a) =
= 0. Then
a∫
0
p(t)|x(t)|λdt ≤ 1
2
a∫
0
[t(a− t)](λ−1)/2p(t)dt
a∫
0
∣∣x′(t)∣∣λ dt. (1.2)
Theorem C. Assume that
(i) l, m, µ and v are nonnegative real numbers such that
1
µ
+
1
v
= 1, and lµ ≥ 1,
(ii) q(t) is a nonnegative and continuous function on [0, a],
(iii) let x1(t) and x2(t) are absolutely continuous functions on [0, a], with x1(0) = x1(a) =
= x2(0) = x2(a) = 0.
Then
*Research is supported by National Natural Science Foundation of China (10971205).
c© C. J. ZHAO, M. BENCZE, 2012
200 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
ON AGARWAL – PANG-TYPE INTEGRAL INEQUALITIES 201
a∫
0
q(t)
[
|x1(t)|l
∣∣x′2(t)∣∣m + |x2(t)|l
∣∣x′1(t)∣∣m] dt ≤
1
2
a∫
0
[t(a− t)](lµ−1)/2qµ(t)dt
1/µ×
×
a∫
0
[
1
µ
(∣∣x′1(t)∣∣lµ +
∣∣x′2(t)∣∣lµ)+
1
v
(∣∣x′1(t)∣∣mv +
∣∣x′2(t)∣∣mv)
]
dt. (1.3)
The main purpose of the present paper is to establish Agarwal – Pang-type inequalities involving
2-order partial derivatives. Our results in special cases yield (1.2) and (1.3), respectively.
Theorem 1.1. Let λ ≥ 1 be a real number, and let p(s, t) be a nonnegative and continuous
functions on [0, a]× [0, b]. Further, let x(s, t) be an absolutely continuous function on [0, a]× [0, b],
with x(s, 0) = x(0, t) = x(0, 0) = 0 and x(a, b) = x(a, t) = x(s, b) = 0. Then
a∫
0
b∫
0
p(s, t)|x(s, t)|λdsdt ≤
≤ 1
2
a∫
0
b∫
0
[st(a− s)(b− t)](λ−1)/2p(s, t)dsdt
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣λ dsdt. (1.4)
Theorem 1.2. Assume that
(i) l, m, µ and v are nonnegative real numbers such that
1
µ
+
1
v
= 1, and lµ ≥ 1,
(ii) q(s, t) is a nonnegative and continuous function on [0, a]× [0, b],
(iii) let j = 1, 2 and xj(s, t) are absolutely continuous functions on [0, a]×[0, b], with xj(s, 0) =
= xj(0, t) = xj(0, 0) = 0 and xj(a, b) = xj(a, t) = xj(s, b) = 0.
Then
a∫
0
b∫
0
q(s, t)
[
|x1(s, t)|l
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣m + |x2(s, t)|l
∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣m] dsdt ≤
≤
1
2
a∫
0
b∫
0
[st(a− s)(b− t)](lµ−1)/2qµ(s, t)dsdt
1/µ×
×
a∫
0
b∫
0
[
1
µ
(∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣lµ +
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣lµ
)
+
+
1
v
(∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣mv +
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣mv)
]
dsdt. (1.5)
We also establish the following Opial-type inequality involving 2-order partial derivatives.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
202 C. J. ZHAO, M. BENCZE
Theorem 1.3. Let j = 1, 2 and λ ≥ 1 be a real number, and let pj(s, t) be a nonnegative and
continuous functions on [0, a] × [0, b]. Further, let xj(s, t) be an absolutely continuous function on
[0, a]× [0, b], with xj(s, 0) = xj(0, t) = xj(0, 0) = 0 and xj(a, b) = xj(a, t) = xj(s, b) = 0. Then
a∫
0
b∫
0
(
p1(s, t)|x1(s, t)|λ + p2(s, t)|x2(s, t)|λ
)
dsdt ≤
≤ 1
2
(
ab
2
)λ−1 [ a∫
0
b∫
0
p1(s, t)dsdt
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣λ dsdt+
+
a∫
0
b∫
0
p2(s, t)dsdt
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣λ dsdt
]
. (1.6)
2. Main results and their proofs.
Theorem 2.1. Let λ ≥ 1 be a real number, and let p(s, t) be a nonnegative and continuous
functions on [0, a]× [0, b]. Further, let x(s, t) be an absolutely continuous function on [0, a]× [0, b],
with x(s, 0) = x(0, t) = x(0, 0) = 0 and x(a, b) = x(a, t) = x(s, b) = 0. Then
a∫
0
b∫
0
p(s, t)|x(s, t)|λdsdt ≤
≤ 1
2
a∫
0
b∫
0
[st(a− s)(b− t)](λ−1)/2p(s, t)dsdt
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣λ dsdt. (2.1)
Proof. From the hypotheses, we have
x(s, t) =
s∫
0
t∫
0
∂2
∂s∂t
x(s, t)dsdt.
By Hölder’s inequality with indices λ and λ/(λ− 1), it follows that
|x(s, t)|λ/2 ≤
s∫
0
t∫
0
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣ dsdt
λ
1/2
≤
≤ (st)(λ−1)/2
s∫
0
t∫
0
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣λ dsdt
1/2 . (2.2)
Similarly, from
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
ON AGARWAL – PANG-TYPE INTEGRAL INEQUALITIES 203
x(s, t) =
a∫
s
b∫
t
∂2
∂s∂t
x(s, t)dsdt,
we obtain
|x(s, t)|λ/2 ≤ [(a− s)(b− t)](λ−1)/2
a∫
s
b∫
t
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣λ dsdt
1/2 . (2.3)
Now a multiplication of (2.2) and (2.3), and by the elementary inequality 2
√
αβ ≤ α + β, α ≥ 0,
β ≤ 0 gives
|x(s, t)|λ ≤ [st(a− s)(b− t)](λ−1)/2
s∫
0
t∫
0
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣λ dsdt
1/2×
×
a∫
s
b∫
t
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣λ dsdt
1/2 ≤
≤ 1
2
[st(a− s)(b− t)](λ−1)/2
s∫
0
t∫
0
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣λ dsdt+
a∫
s
b∫
t
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣λ dsdt
=
=
1
2
[st(a− s)(b− t)](λ−1)/2
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣λ dsdt. (2.4)
Multiplying the both sides of (2.4) by p(s, t) and integrating both sides over t from 0 to b first and
then integrating the resulting inequality over s from 0 to a, we obtain
a∫
0
b∫
0
p(s, t)|x(s, t)|λdsdt ≤
≤ 1
2
a∫
0
b∫
0
[st(a− s)(b− t)](λ−1)/2p(s, t)
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣λ dsdt
dsdt =
=
1
2
a∫
0
b∫
0
[st(a− s)(b− t)](λ−1)/2p(s, t)dsdt
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣λ dsdt.
Remark 2.1. Let x(s, t) reduce to s(t) and with suitable modifications, Theorem 2.1 becomes
Theorem B stated in the introduction which was given by Agarwal and Pang [17].
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
204 C. J. ZHAO, M. BENCZE
Remark 2.2. Taking for p(s, t) = constant in (2.1), we have
a∫
0
b∫
0
|x(s, t)|λdsdt ≤ 1
2
(ab)λ
[
B
(
λ+ 1
2
,
λ+ 1
2
)]2 a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣λ dsdt,
where B is the Beta function.
Theorem 2.2. Let j = 1, 2 and λ ≥ 1 be a real number, and let pj(s, t) be a nonnegative and
continuous functions on [0, a] × [0, b]. Further, let xj(s, t) be an absolutely continuous function on
[0, a]× [0, b], with xj(s, 0) = xj(0, t) = xj(0, 0) = 0 and xj(a, b) = xj(a, t) = xj(s, b) = 0. Then
a∫
0
b∫
0
(
p1(s, t)|x1(s, t)|λ + p2(s, t)|x2(s, t)|λ
)
dsdt ≤
≤ 1
2
(
ab
2
)λ−1 [ a∫
0
b∫
0
p1(s, t)dsdt
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣λ dsdt+
+
a∫
0
b∫
0
p2(s, t)dsdt
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣λ dsdt
]
. (2.5)
Proof. From the hypotheses, we have
x1(s, t) =
s∫
0
t∫
0
∂2
∂s∂t
x1(s, t)dsdt and x1(s, t) =
a∫
s
b∫
t
∂2
∂s∂t
x1(s, t)dsdt.
Hence
|x1(s, t)| ≤
1
2
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣ dsdt.
By Hölder’s inequality with indices λ and λ/(λ− 1), it follows that
p1(s, t)|x1(s, t)|λ ≤
1
2λ
p1(s, t)
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣ dsdt
λ ≤
≤ 1
2
(
ab
2
)λ−1
p1(s, t)
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣λ dsdt. (2.6)
Similarly
p2(s, t)|x2(s, t)|λ ≤
1
2λ
p2(s, t)
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣ dsdt
λ ≤
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
ON AGARWAL – PANG-TYPE INTEGRAL INEQUALITIES 205
≤ 1
2
(
ab
2
)λ−1
p2(s, t)
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣λ dsdt. (2.7)
Taking the sum of (2.6) and (2.7) and integrating the resulting inequalities over t from 0 to b first
and then over s from 0 to a, we obtain (2.5).
Remark 2.3. Taking for x1(s, t) = x1(s, t) = x(s, t) in (2.5), (2.5) changes to the following
inequality:
a∫
0
b∫
0
p(s, t)|x(s, t)|λdsdt ≤
≤ 1
2
(
ab
2
)λ−1 a∫
0
b∫
0
p(s, t)dsdt
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣λ dsdt. (2.8)
Let x(s, t) reduce to s(t) and with suitable modifications, (2.8) becomes the following result:
a∫
0
p(t)|x(t)|λdt ≤ 1
2
(a
2
)λ−1 a∫
0
p(t)dt
a∫
0
∣∣x′(t)∣∣λ dt. (2.9)
This is just a new inequality established by Agarwal and Pang [17]. For λ = 2 the inequality (2.9)
has appear in the work of Traple [18], Pachpatte [19] proved it for λ = 2m (m ≥ 1 an integer).
Remark 2.4. Let xj(s, t) reduce to sj(t), j = 1, 2, and with suitable modifications, (2.5) be-
comes the following interesting result:
a∫
0
(
p1(t)|x1(t)|λ + p2(t)|x2(t)|λ
)
dt ≤
≤ 1
2
(a
2
)λ−1 [ a∫
0
p1(t)dt
a∫
0
∣∣x′1(t)∣∣λ dt+
a∫
0
p2(t)dt
a∫
0
∣∣x′2(t)∣∣λ dt
]
.
Theorem 2.3. Assume that
(i) l,m, µ and v are nonnegative real numbers such that
1
µ
+
1
v
= 1, and lµ ≥ 1,
(ii) q(s, t) is a nonnegative and continuous function on [0, a]× [0, b],
(iii) let j = 1, 2 and xj(s, t) are absolutely continuous functions on [0, a]×[0, b], with xj(s, 0) =
= xj(0, t) = xj(0, 0) = 0 and xj(a, b) = xj(a, t) = xj(s, b) = 0.
Then
a∫
0
b∫
0
q(s, t)
[
|x1(s, t)|l
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣m + |x2(s, t)|l
∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣m] dsdt ≤
≤
1
2
a∫
0
b∫
0
[st(a− s)(b− t)](lµ−1)/2qµ(s, t)dsdt
1/µ×
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
206 C. J. ZHAO, M. BENCZE
×
a∫
0
b∫
0
[
1
µ
(∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣lµ +
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣lµ
)
+
+
1
v
(∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣mv +
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣mv)
]
dsdt. (2.10)
Proof. From the Hölder’s inequality, the inequality (2.1) and Young’s inequality wz ≤ wµ
µ
+
zv
v
we have
a∫
0
b∫
0
q(s, t)|x1(s, t)|l
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣m dsdt ≤
≤
a∫
0
b∫
0
qµ(s, t)|x1(s, t)|lµdsdt
1/µ a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣mv dsdt
1/v ≤
≤
1
2
a∫
0
b∫
0
[st(a− s)(b− t)](lµ−1)/2qµ(s, t)dsdt
1/µ a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x(s, t)
∣∣∣∣lµ dsdt
1/µ×
×
a∫
0
b∫
0
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣mv dsdt
1/v ≤
≤
1
2
a∫
0
b∫
0
[st(a− s)(b− t)](lµ−1)/2qµ(s, t)dsdt
1/µ×
×
a∫
0
b∫
0
(
1
µ
∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣lµ +
1
v
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣mv
)
dsdt, (2.11)
and similarly
a∫
0
b∫
0
q(s, t)|x1(s, t)|m
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣l dsdt ≤
≤
1
2
a∫
0
b∫
0
[st(a− s)(b− t)](lµ−1)/2qµ(s, t)dsdt
1/µ×
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
ON AGARWAL – PANG-TYPE INTEGRAL INEQUALITIES 207
×
a∫
0
b∫
0
(
1
v
∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣mv +
1
µ
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣lµ
)
dsdt. (2.12)
An addition of (2.11) and (2.12) gives the inequality (2.10).
Remark 2.5. Let x(s, t) reduce to s(t) and with suitable modifications, (2.10) becomes the
following result:
a∫
0
q(t)
[
|x1(t)|l
∣∣x′2(t)∣∣m + |x2(t)|l
∣∣x′1(t)∣∣m] dt ≤
≤
1
2
a∫
0
[t(a− t)](lµ−1)/2qµ(t)dt
1/µ×
×
a∫
0
[
1
µ
(∣∣x′1(t)∣∣lµ +
∣∣x′2(t)∣∣lµ)+
1
v
(∣∣x′1(t)∣∣mv +
∣∣x′2(t)∣∣mv)
]
dt. (2.13)
The inequality (2.13) has appeared in the work of Agarwal and Pang [17].
Remark 2.6. Let q(s, t) 6= constant and taking for µ = v = 2 in (2.10), we obtain
a∫
0
b∫
0
q(s, t)
(
|x1(s, t)|l
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣m + |x2(s, t)|l
∣∣∣∣ ∂1∂s∂t
x1(s, t)
∣∣∣∣m) dsdt ≤
≤
1
8
a∫
0
b∫
0
[st(a− s)(b− t)](2l−1)/2q2(s, t)dsdt
1/2×
×
a∫
0
b∫
0
(∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣2l +
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣2l +
+
∣∣∣∣ ∂2∂s∂t
x1(s, t)
∣∣∣∣2m +
∣∣∣∣ ∂2∂s∂t
x2(s, t)
∣∣∣∣2m
)
dsdt. (2.14)
Let xj(s, t), j = 1, 2, reduce to sj(t) and with suitable modifications, (2.14) becomes the following
result:
a∫
0
q(t)
(
|x1(t)|l
∣∣x′2(t)∣∣m + |x2(t)|l
∣∣x′1(t)∣∣m) dt ≤
≤
1
8
a∫
0
[t(a− t)](2l−1)/2q2(t)dt
1/2×
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
208 C. J. ZHAO, M. BENCZE
×
a∫
0
(∣∣x′1(t)∣∣2l +
∣∣x′2(t)∣∣2l +
∣∣x′1(t)∣∣2m +
∣∣x′2(t)∣∣2m) dt. (2.15)
This is just an inequality established by Agarwal and Pang [17]. The inequality (2.15) is sharper than
the following inequality (see [17]):
a∫
0
q(t)
(
|x1(t)|l
∣∣x′2(t)∣∣m + |x2(t)|l
∣∣x′1(t)∣∣m) dt ≤
≤
h2l−1
4l+1
a∫
0
q2(t)dt
1/2 a∫
0
(∣∣x′1(t)∣∣2l +
∣∣x′2(t)∣∣2l +
∣∣x′1(t)∣∣2m +
∣∣x′2(t)∣∣2m) dt.
For the integers l,m ≥ 1, the inequality (2.15) has been obtained by Lin [20].
3. Uniqueness of initial value problem. Here, as application to one of the inequalities obtained
in Section 2 we shall prove the uniqueness of the solution of initial value problem involving higher
order ordinary equation.
Theorem 3.1. For the system of differential equations
y
′′
j = fj(t, y1, y
′
1, y2, y
′
2), j = 1, 2, (3.1)
together with the initial conditions
y
(i)
j (0) = yj,i, j = 1, 2, 0 ≤ i ≤ 1, (3.2)
we assume that fj : [0, τ ]× R2 × R2 → R are continuous, and satisfy the Lipschitz condition
|fj (t, y1,0, y1,1, y2,0, y2,1)− fj (t, ȳ1,0, ȳ1,1, ȳ2,0, ȳ2,1)| ≤
≤
1∑
k=0
[
q1,j,k(t)|y1,k − ȳ1,k|+ q2,j,k(t)|y2,k − ȳ2,k|
]
,
where the functions qr,j,k ≥ 0, 1 ≤ r, 4j ≤ 2, 0 ≤ k ≤ 1, are continuous on [0, τ ]. Then the
problem (3.1), (3.2) has at most one solution on [0, τ ].
Proof. If the problem (3.1), (3.2) has two solutions (y1(t), y2(t)), (ȳ1(t), ȳ2(t)) then for the
functions xj(t) = yj(t)− ȳj(t), j = 1, 2, it follows
|x′′
j (t)|2 ≤
1∑
k=0
[
q1,j,k(t)|x
(k)
1 (t)||x′′
j (t)|+ q2,j,k(t)|x
(k)
2 (t)||x′′
j (t)|
]
.
Summing these two inequalities, and integrating from 0 to t, we obtain
t∫
0
[
|x′′
1(s)|2 + |x′′
2(s)|2
]
ds ≤
1∑
k=0
t∫
0
q̄k(s)
[
|x(k)1 (s)||x′′
1(s)|+ |x(k)2 (s)||x′′
2(s)|
]
ds+
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
ON AGARWAL – PANG-TYPE INTEGRAL INEQUALITIES 209
+
1∑
k=0
t∫
0
q̄∗k(s)
[
|x(k)2 (s)||x′′
2(s)|+ |x(k)1 (s)||x′′
1(s)|
]
ds, (3.3)
where q̄k(t) = max(q1,1,k(t), q2,2,k(t)) and q∗k(t) = max(q2,1,k(t), q1,2,k(t)).
For 0 ≤ k ≤ 1, on the right side of (3.3), we apply the inequality (2.10) with x(s, t) = x(t),
l = 1, µ = 2,m = 1 and with suitable modifications, to obtain the following inequality:
t∫
0
[
|x′′
1(s)|2 + |x′′
2(s)|2
]
ds ≤ K(t)
t∫
0
[
|x′′
1(s)|2 + |x′′
2(s)|2
]
ds, (3.4)
where K(t) is a continuous function with the property K(0) = 0. Hence, the inequality (3.4) implies
that y1(t) = ȳ1(t), y2(t) = ȳ2(t) and t ∈ [0, τ ].
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Received 25.02.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
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| id | umjimathkievua-article-2566 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:56Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4d/653f88d2f8b093d7d52109406b08d84d.pdf |
| spelling | umjimathkievua-article-25662020-03-18T19:29:46Z On Agarwal - Pang-type integral inequalities Про нерiвностi типу Агарвала – Панга Bencze, M. Zhao, C. J Бенче, М. Чжао, С. Дж. We establish some new Agarwal – Pang-type inequalities involving second-order partial derivatives. Our results in special cases yield some of interrelated results and provide new estimates for inequalities of this type. Встановлено деякi новi нерiвностi типу Агарвала – Панга, що мiстять частиннi похiднi другого порядку. В окремих випадках iз одержаних результатiв випливають деякi пов’язанi результати та новi оцiнки для нерiвностей цього типу. Institute of Mathematics, NAS of Ukraine 2012-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2566 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 2 (2012); 199-209 Український математичний журнал; Том 64 № 2 (2012); 199-209 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2566/1891 https://umj.imath.kiev.ua/index.php/umj/article/view/2566/1892 Copyright (c) 2012 Bencze M.; Zhao C. J |
| spellingShingle | Bencze, M. Zhao, C. J Бенче, М. Чжао, С. Дж. On Agarwal - Pang-type integral inequalities |
| title | On Agarwal - Pang-type integral inequalities |
| title_alt | Про нерiвностi типу Агарвала – Панга |
| title_full | On Agarwal - Pang-type integral inequalities |
| title_fullStr | On Agarwal - Pang-type integral inequalities |
| title_full_unstemmed | On Agarwal - Pang-type integral inequalities |
| title_short | On Agarwal - Pang-type integral inequalities |
| title_sort | on agarwal - pang-type integral inequalities |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2566 |
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