Lyapunov-Type Inequalities for Quasilinear Systems with Antiperiodic Boundary Conditions
We establish some new Lyapunov-type inequalities for one-dimensional p-Laplacian systems with antiperiodic boundary conditions. The lower bounds of eigenvalues are presented.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508455106772992 |
|---|---|
| author | Bai, Yongzhen Li, Yannan Wang, Youyu Бай, Йонгжен Лі, Янань Ван, Й. |
| author_facet | Bai, Yongzhen Li, Yannan Wang, Youyu Бай, Йонгжен Лі, Янань Ван, Й. |
| author_sort | Bai, Yongzhen |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:26:06Z |
| description | We establish some new Lyapunov-type inequalities for one-dimensional p-Laplacian systems with antiperiodic boundary conditions. The lower bounds of eigenvalues are presented. |
| first_indexed | 2026-03-24T02:25:28Z |
| format | Article |
| fulltext |
UDC 517.9
Youyu Wang, Yannan Li, Yongzhen Bai (Tianjin Univ. Finance and Economics, China)
LYAPUNOV-TYPE INEQUALITIES FOR QUASILINEAR SYSTEMS
WITH ANTIPERIODIC BOUNDARY CONDITIONS
НЕРIВНОСТI ТИПУ ЛЯПУНОВА ДЛЯ КВАЗIЛIНIЙНИХ СИСТЕМ
З АНТИПЕРIОДИЧНИМИ ГРАНИЧНИМИ УМОВАМИ
We establish some new Lyapunov-type inequalities for one-dimensional p-Laplacian systems with antiperiodic boundary
conditions. The lower bounds of eigenvalues are presented.
Встановлено деякi новi нерiвностi типу Ляпунова для одновимiрних p-лапласових систем з антиперiодичними
граничними умовами. Наведено нижнi межi для власних значень.
1. Introduction. The Lyapunov inequality and many of its generalizations have proved to be useful
tools in oscillation theory, disconjugacy, eigenvalue problems, and numerous other applications for
the theories of differential and difference equations. A classical result of Lyapunov [12] states that if
u(t) is a nontrivial solution of the differential system
u′′(t) + r(t)u(t) = 0, t ∈ (a, b),
u(a) = 0 = u(b),
where r(t) is a continuous and nonnegative function defined in [a, b], then
b∫
a
r(t)dt >
4
b− a
,
and the constant 4 cannot be replaced by a larger number.
Since the appearance of Lyapunov’s fundamental paper, various proofs and generalizations or
improvements have appeared in the literature. For authors, who contributed to the Lyapunov-type
inequalities, we refer to [1, 3 – 11, 14 – 16, 20 – 23] and the references quoted therein, especially to
the survey papers [1, 3, 22]. In recent years, the Lyapunov inequality has been extended in many
directions.
For example, in 2004, Pinasco [18] has generalized the classical Lyapunov inequality for the
half-linear differential equation
(|u′(t)|p−2u′(t))′ + r(t)|u(t)|p−2u(t) = 0, t ∈ (a, b),
u(a) = u(b) = 0.
(1.1)
He obtained Lyapunov inequality for system (1.1) as follows:
b∫
a
r(t)dt ≥ 2p
(b− a)p/q
,
c© YOUYU WANG, YANNAN LI, YONGZHEN BAI, 2013
1646 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
LYAPUNOV-TYPE INEQUALITIES FOR QUASILINEAR SYSTEMS WITH ANTIPERIODIC . . . 1647
where p > 1, 1/p+ 1/q = 1 and r ∈ C([a, b], (0,+∞)).
In 2006, Pinasco [17] studied the following Dirichlet – Neumann problem:
(|u′(t)|p−2u′(t))′ + r(t)|u(t)|p−2u(t) = 0, t ∈ (a, b),
u′(a) = u(b) = 0.
He gave the following Lyapunov-type inequality:
b∫
a
r(t)dt >
1
(b− a)p−1
.
Although there is an extensive literature on the Lyapunov-type inequalities for various classes of
differential equations, there is not much done for the linear Hamiltonian systems.
In 2006, Napoli and Pinasco [13] have interested in the problem of finding the Lyapunov-type
inequality for the following quasilinear systems:
− (|u′1(t)|p1−2u′1(t))′ = r1(t)|u1(t)|α1−2|u2(t)|α2u1(t),
− (|u′2(t)|p2−2u′2(t))′ = r2(t)|u1(t)|α1 |u2(t)|α2−2u2(t)
(1.2)
with the Dirichlet boundary conditions
u1(a) = u1(b) = 0 = u2(a) = u2(b), u1(t) > 0, u2(t) > 0 ∀t ∈ (a, b),
where r1, r2 are real-valued positive continuous functions for all x ∈ R, the exponents satisfy
1 < p1, p2 <∞, and the positive parameters α1, α2 satisfy
α1
p1
+
α2
p2
= 1. They gave the following
Lyapunov-type inequality:
(b− a)α1+α2−1
b∫
a
r1(t)dt
α1
p1
b∫
a
r2(t)dt
α2
p2
≥ 2α1+α2 . (1.3)
More recently, by adopting the method used in Napoli and Pinasco [13], Cakmak and Tiryaki [2]
generalized Lyapunov-type inequality (1.3) to the following more general quasilinear systems:
− (|u′1(t)|p1−2u′1(t))′ = r1(t)|u1(t)|α1−2|u2(t)|α2 . . . |un(t)|αnu1(t),
− (|u′2(t)|p2−2u′2(t))′ = r2(t)|u1(t)|α1 |u2(t)|α2−2 . . . |un(t)|αnu2(t),
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
− (|u′n(t)|pn−2u′n(t))′ = rn(t)|u1(t)|α1 |u2(t)|α2 . . . |un(t)|αn−2un(t)
(1.4)
with Dirichlet boundary conditions
ui(a) = 0 = ui(b), ui(t) 6≡ 0 ∀t ∈ (a, b), i = 1, 2, . . . , n.
They established the Lyapunov-type inequality
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
1648 YOUYU WANG, YANNAN LI, YONGZHEN BAI
n∏
i=1
b∫
a
r+i (t)dt
αi
pi
≥ 2m(b− a)1−m,
where m =
∑n
i=1
αi and r+i (t) = max{ri(t), 0} for i = 1, 2, · · · , n.
Motivated by the paper [2] and [19], the purpose of this paper is to get three types of Lyapunov
inequalities for one-dimensional p-Laplacian system. In Section 2, we show Lyapunov inequality for
one-dimensional p-Laplacian problem
− (|u′(t)|p−2u′(t))′ = r(t)|u(t)|p−2u(t), t ∈ (a, b),
u(a) + u(b) = 0, u′(a) + u′(b) = 0,
(1.5)
where r : [a, b] → (0,∞) is a continuous function, p > 1, q be a conjugate exponent of p, i.e.,
1/p+ 1/q = 1.
In Section 3, the more general system than (1.2)
− (|u′1(t)|p1−2u′1(t))′ = r1(t)|u1(t)|α1−2|u2(t)|α2u1(t),
− (|u′2(t)|p2−2u′2(t))′ = r2(t)|u1(t)|β1 |u2(t)|β2−2u2(t),
ui(a) + ui(b) = 0, u′i(a) + u′i(b) = 0, i = 1, 2,
(1.6)
will be studied, we establish new Lyapunov inequality for this system. Where r1, r2 are real-valued
positive continuous functions for all t ∈ R, the exponents satisfy 1 < p1, p2 <∞, α1, α2, β1, β2 > 0
satisfy
α1
p1
+
α2
p2
= 1 and
β1
p1
+
β2
p2
= 1.
In Section 4, we consider the Lyapunov inequality for system (1.4) with antiperiodic conditions,
i.e.,
− (|u′1(t)|p1−2u′1(t))′ = r1(t)|u1(t)|α1−2|u2(t)|α2 . . . |un(t)|αnu1(t),
− (|u′2(t)|p2−2u′2(t))′ = r2(t)|u1(t)|α1 |u2(t)|α2−2 . . . |un(t)|αnu2(t),
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
− (|u′n(t)|pn−2u′n(t))′ = rn(t)|u1(t)|α1 |u2(t)|α2 . . . |un(t)|αn−2un(t),
ui(a) + ui(b) = 0, u′i(a) + u′i(b) = 0, i = 1, 2, . . . , n.
(1.7)
As an application of our Lyapunov-type inequalities, in Section 5, we focus on the estimate of
lower bound for eigenvalues.
2. Lyapunov-type inequality for system (1.5). In the proof of our results, the following lemma
is very important.
Lemma 2.1. If u′ ∈ Lp(a, b) and u(t) satisfying the condition u(a) + u(b) = 0, then
|u(t)| ≤ 1
2
(b− a)1/q
b∫
a
|u′(x)|pdx
1/p
.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
LYAPUNOV-TYPE INEQUALITIES FOR QUASILINEAR SYSTEMS WITH ANTIPERIODIC . . . 1649
Proof. Let us define,
H(x, y) =
1
2
, a ≤ x ≤ y,
−1
2
, y ≤ x ≤ b.
Then, we have
u(y) =
b∫
a
u′(x)H(x, y)dx, a ≤ y ≤ b.
So, by Hölder’s inequality, we have
|u(y)| ≤
b∫
a
|H(x, y)|qdx
1/q b∫
a
|u′(x)|pdx
1/p
=
=
1
2
(b− a)1/q
b∫
a
|u′(x)|pdx
1/p
.
The main result of this section is the following Lyapunov-type inequality.
Theorem 2.1. If u(t) is a solution of system (1.5) and u(t) 6≡ 0, t ∈ [a, b], then
b∫
a
r(t)dt >
2p
(b− a)p−1
.
Proof. Multiplying the equation in (1.5) by u and integrating over [a, b], yields
−
b∫
a
(|u′|p−2u′)′udt =
b∫
a
r|u|pdt,
|u′(a)|p−2u′(a)u(a)− |u′(b)|p−2u′(b)u(b) +
b∫
a
|u′|pdt =
b∫
a
r|u|pdt,
by the antiperiodic boundary condition in (1.5), we get |u′(b)|p−2u′(b)u(b) = |u′(a)|p−2u′(a)u(a),
so we have
b∫
a
|u′|pdt =
b∫
a
r|u|pdt,
by Lemma 2.1, we obtain
b∫
a
|u′|pdt =
b∫
a
r|u|pdt < ( max
a≤t≤b
|u|)p
b∫
a
r(t)dt ≤ (b− a)p−1
2p
b∫
a
|u′(t)|pdt
b∫
a
r(t)dt. (2.1)
Now we claim that
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
1650 YOUYU WANG, YANNAN LI, YONGZHEN BAI
b∫
a
|u′(t)|pdt > 0. (2.2)
If not, we have
∫ b
a
|u′(t)|pdt = 0, then u′(t) = 0, t ∈ [a, b]. By the antiperiodic boundary condition
u(a) + u(b) = 0, we can obtain u(t) ≡ 0, t ∈ [a, b], which contradicts u(t) 6≡ 0, t ∈ [a, b]. Therefore
(2.2) holds. Divided the inequality (2.1) by
∫ b
a
|u′(t)|pdt, we have
b∫
a
r(t)dt >
2p
(b− a)p−1
.
3. Lyapunov-type inequality for system (1.6). In this section, the main result is as follows.
Theorem 3.1. If system (1.6) has a solution u1(t), u2(t) and ui(t) 6≡ 0 ∀t ∈ [a, b], i = 1, 2,
then
(b− a)β1+α2−β1p1−
α2
p2
b∫
a
r1(t)dt
β1
p1
b∫
a
r2(t)dt
α2
p2
≥ 2β1+α2 . (3.1)
Proof. Multiplying the first equation in (1.6) by u1 and integrating over [a, b] together with the
antiperiodic condition, yields
b∫
a
|u′1(t)|p1dt =
b∫
a
r1(t)|u1(t)|α1 |u2(t)|α2dt, (3.2)
similarly,
b∫
a
|u′2(t)|p2dt =
b∫
a
r2(t)|u1(t)|β1 |u2(t)|β2dt, (3.3)
by Lemma 2.1, we get
|u1(t)|p1 ≤
(b− a)p1−1
2p1
b∫
a
|u′1(x)|p1dx. (3.4)
Now, it follows from (3.2), (3.4) and the Hölder inequality that
b∫
a
r1(t)|u1(t)|p1dt ≤
(b− a)p1−1
2p1
b∫
a
r1(t)dt
b∫
a
|u′1(t)|p1dt =
=
(b− a)p1−1
2p1
b∫
a
r1(t)dt
b∫
a
r1(t)|u1(t)|α1 |u2(t)|α2dt ≤
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
LYAPUNOV-TYPE INEQUALITIES FOR QUASILINEAR SYSTEMS WITH ANTIPERIODIC . . . 1651
≤ (b− a)p1−1
2p1
b∫
a
r1(t)dt
b∫
a
r1(t)|u1(t)|p1dt
α1
p1
b∫
a
r1(t)|u2(t)|p2dt
α2
p2
=
=M1
b∫
a
r1(t)|u1(t)|p1dt
α1
p1
b∫
a
r1(t)|u2(t)|p2dt
α2
p2
(3.5)
and
b∫
a
r2(t)|u1(t)|p1dt ≤
(b− a)p1−1
2p1
b∫
a
r2(t)dt
b∫
a
|u′1(t)|p1dt =
=
(b− a)p1−1
2p1
b∫
a
r2(t)dt
b∫
a
r1(t)|u1(t)|α1 |u2(t)|α2dt ≤
≤ (b− a)p1−1
2p1
b∫
a
r2(t)dt
b∫
a
r1(t)|u1(t)|p1dt
α1
p1
b∫
a
r1(t)|u2(t)|p2dt
α2
p2
=
=M2
b∫
a
r1(t)|u1(t)|p1dt
α1
p1
b∫
a
r1(t)|u2(t)|p2dt
α2
p2
, (3.6)
where
M1 =
(b− a)p1−1
2p1
b∫
a
r1(t)dt, M2 =
(b− a)p1−1
2p1
b∫
a
r2(t)dt. (3.7)
Similarly, we also have
|u2(t)|p2 ≤
(b− a)p2−1
2p2
b∫
a
|u′2(x)|p2dx. (3.8)
It follows from (3.3), (3.8) and the Hölder inequality that
b∫
a
r1(t)|u2(t)|p2dt ≤
(b− a)p2−1
2p2
b∫
a
r1(t)dt
b∫
a
|u′2(t)|p2dt =
=
(b− a)p2−1
2p2
b∫
a
r1(t)dt
b∫
a
r2(t)|u1(t)|β1 |u2(t)|β2dt ≤
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
1652 YOUYU WANG, YANNAN LI, YONGZHEN BAI
≤ (b− a)p2−1
2p2
b∫
a
r1(t)dt
b∫
a
r2(t)|u1(t)|p1dt
β1
p1
b∫
a
r2(t)|u2(t)|p2dt
β2
p2
=
=M3
b∫
a
r2(t)|u1(t)|p1dt
β1
p1
b∫
a
r2(t)|u2(t)|p2dt
β2
p2
(3.9)
and
b∫
a
r2(t)|u2(t)|p2dt ≤
(b− a)p2−1
2p2
b∫
a
r2(t)dt
b∫
a
|u′2(t)|p2dt =
=
(b− a)p2−1
2p2
b∫
a
r2(t)dt
b∫
a
r2(t)|u1(t)|β1 |u2(t)|β2dt ≤
≤ (b− a)p2−1
2p2
b∫
a
r2(t)dt
b∫
a
r2(t)|u1(t)|p1dt
β1
p1
b∫
a
r2(t)|u2(t)|p2dt
β2
p2
=
=M4
b∫
a
r2(t)|u1(t)|p1dt
β1
p1
b∫
a
r2(t)|u2(t)|p2dt
β2
p2
, (3.10)
where
M3 =
(b− a)p2−1
2p2
b∫
a
r1(t)dt, M4 =
(b− a)p2−1
2p2
b∫
a
r2(t)dt. (3.11)
Next, we prove that
b∫
a
r1(t)|u1(t)|p1dt > 0. (3.12)
In fact, if (3.12) is not true, then
b∫
a
r1(t)|u1(t)|p1dt = 0. (3.13)
From (3.2) and (3.13), we have
0 ≤
b∫
a
|u′1(t)|p1dt =
b∫
a
r1(t)|u1(t)|α1 |u2(t)|α2dt ≤
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
LYAPUNOV-TYPE INEQUALITIES FOR QUASILINEAR SYSTEMS WITH ANTIPERIODIC . . . 1653
≤
b∫
a
r1(t)|u1(t)|p1dt
α1
p1
b∫
a
r1(t)|u2(t)|p2dt
α2
p2
= 0.
So u′1(t) ≡ 0, a ≤ t ≤ b, by the antiperiodic boundary condition, we obtain u1(t) ≡ 0 for a ≤ t ≤ b,
which contradicts u1(t) 6≡ 0 ∀t ∈ [a, b]. Therefore, (3.12) holds. Similarly, we get
b∫
a
r1(t)|u2(t)|p2dt > 0,
b∫
a
r2(t)|u2(t)|p2dt > 0,
b∫
a
r2(t)|u1(t)|p1dt > 0.
From (3.5), (3.6), (3.9), (3.10), we obtain
M
α1β1
p21
1 M
β1α2
p1p2
2 M
β1α2
p1p2
3 M
α2β2
p22
4 ≥ 1.
It follows from (3.7), (3.11) that (3.1) holds.
4. Lyapunov-type inequality for system (1.7). In this section, we establish new Lyapunov-type
inequality for system (1.7). Assume that
(H1) ri, i = 1, 2, . . . , n, are real-valued positive continuous functions for all t ∈ R,
(H2) the exponents satisfy 1 < pi <∞ and the positive parameters αi satisfy
∑n
i=1
αi
pi
= 1.
Theorem 4.1. If system (1.7) has a solution (u1(t), u2(t), . . . , un(t)) with ui(t) 6≡ 0, t ∈ [a, b],
i = 1, 2, . . . , n, then
n∏
i=1
n∏
j=1
(b− a)pi−1
2pi
b∫
a
rj(t)dt
αiαj
pipj
≥ 1. (4.1)
Proof. Multiplying the ith equation in (1.7) by ui and integrating over [a, b] together with the
antiperiodic condition, yields
b∫
a
|u′i(t)|pidt =
b∫
a
ri(t)
n∏
k=1
|uk(t)|αkdt, i = 1, 2, . . . , n, (4.2)
similarly, by Lemma 2.1, we get
|ui(t)|pi ≤
(b− a)pi−1
2pi
b∫
a
|u′i(x)|pidx, i = 1, 2, . . . , n. (4.3)
Now, it follows from (4.2), (4.3) and the generalized Hölder inequality that
b∫
a
rj(t)|ui(t)|pidt ≤
(b− a)pi−1
2pi
b∫
a
rj(t)dt
b∫
a
|u′i(t)|pidt =
=
(b− a)pi−1
2pi
b∫
a
rj(t)dt
b∫
a
ri(t)
n∏
k=1
|uk(t)|αkdt =
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
1654 YOUYU WANG, YANNAN LI, YONGZHEN BAI
=Mij
b∫
a
ri(t)
n∏
k=1
|uk(t)|αkdt ≤Mij
n∏
k=1
b∫
a
ri(t)|uk(t)|pkdt
αk
pk
, i, j = 1, 2, . . . , n, (4.4)
where
Mij =
(b− a)pi−1
2pi
b∫
a
rj(t)dt, i, j = 1, 2, . . . , n. (4.5)
Next, we prove that
b∫
a
ri(t)|uj(t)|pjdt > 0, i, j = 1, 2, . . . , n. (4.6)
If (4.6) is not true, then there exists i0, j0 ∈ {1, 2, . . . , n} such that
b∫
a
ri0(t)|uj0(t)|pj0dt = 0. (4.7)
From (4.2) and (4.7), we have
0 ≤
b∫
a
|u′i0(t)|
pi0dt =
b∫
a
ri0(t)
n∏
k=1
|uk(t)|αkdt ≤
n∏
k=1
b∫
a
ri0(t)|uk(t)|pkdt
αk
pk
= 0.
So that
u′i0(t) = 0, a ≤ t ≤ b. (4.8)
Combining the antiperiodic boundary and (4.8), we obtain that ui0(t) ≡ 0 for a ≤ t ≤ b, which
contradicts ui(t) 6≡ 0, t ∈ [a, b], i = 1, 2, . . . , n. Therefore (4.6) holds. From (4.4), we have
n∏
i=1
n∏
j=1
M
αiαj
pipj
ij ≥ 1.
It follows from (4.5) that (4.1) holds.
5. Lower bounds for eigenvalues problem. In this section, we apply our Lyapunov-type in-
equalities to obtain lower bounds for eigenvalues. Firstly, we investigate the problem
− (|u′(t)|p−2u′(t))′ = λr(t)|u(t)|p−2u(t), t ∈ (a, b),
u(a) + u(b) = 0, u′(a) + u′(b) = 0.
(5.1)
As a corollary of Theorem 2.1, we have the result for system (5.1).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
LYAPUNOV-TYPE INEQUALITIES FOR QUASILINEAR SYSTEMS WITH ANTIPERIODIC . . . 1655
Theorem 5.1. Assume that p > 1, r(t) is real-valued positive continuous function for all t ∈ R
and system (5.1) has a solution u(t) satisfying u(t) 6≡ 0, t ∈ [a, b]. Let λ be the eigenvalue of system
(5.1). Then
λ >
2p
(b− a)p−1
∫ b
a
r(x)dx
.
Secondly, we consider the eigenvalues problem
− (|u′1(t)|p1−2u′1(t))′ = λ1α1r(t)|u1(t)|α1−2|u2(t)|α2 . . . |un(t)|αnu1(t),
− (|u′2(t)|p2−2u′2(t))′ = λ2α2r(t)|u1(t)|α1 |u2(t)|α2−2 . . . |un(t)|αnu2(t),
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
− (|u′n(t)|pn−2u′n(t))′ = λnαnr(t)|u1(t)|α1 |u2(t)|α2 . . . |un(t)|αn−2un(t),
ui(a) + ui(b) = 0, u′i(a) + u′i(b) = 0, i = 1, 2, . . . , n.
(5.2)
Let (λ1, λ2, . . . , λn) be eigenvalue of problem (5.2) and (u1(t), u2(t), . . . , un(t)) be the eigen-
functions associated with (λ1, λ2, . . . , λn). Then (u1(t), u2(t), . . . , un(t)) is a solution of system
(1.7) with ri(t) = λiαir(t) > 0 for i = 1, 2, . . . , n.
Theorem 5.2. Assume that 1 < pi < ∞, αi > 0 satisfy
∑n
i=1
αi
pi
= 1, r(t) is a real-valued
positive continuous function defined on R, system (5.2) has a solution (u1(t), u2(t), . . . , un(t))
with ui(t) 6≡ 0, t ∈ [a, b], i = 1, 2, . . . , n. Then there exists a function g(λ1, λ2, . . . , λn−1)
such that λn > g(λ1, λ2, . . . , λn−1) for every eigenvalue (λ1, λ2, . . . , λn) of system (5.2), where
g(λ1, λ2, . . . , λn−1) is given by
g(λ1, λ2, . . . , λn−1) =
1
αn
n−1∏
j=1
(λjαj)
αj
pj
n∏
i=1
(b− a)pi−1
2pi
b∫
a
rj(t)dt
αi
pi
− pn
αn
.
Proof. For system (1.7), ri(t) = λiαir(t) > 0 for i = 1, 2, . . . , n. Hence, it follows from (4.1)
that
1 ≤
n∏
i=1
n∏
j=1
(b− a)pi−1
2pi
b∫
a
rj(t)dt
αiαj
pipj
=
=
n∏
j=1
(λjαj)
αj
pj
n∏
i=1
(b− a)pi−1
2pi
b∫
a
rj(t)dt
αi
pi
=
= (λnαn)
αn
pn
n−1∏
j=1
(λjαj)
αj
pj
n∏
i=1
(b− a)pi−1
2pi
b∫
a
rj(t)dt
αi
pi
.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
1656 YOUYU WANG, YANNAN LI, YONGZHEN BAI
Hence, we have
λn >
1
αn
n−1∏
j=1
(λjαj)
αj
pj
n∏
i=1
(b− a)pi−1
2pi
b∫
a
rj(t)dt
αi
pi
− pn
αn
.
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ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
|
| id | umjimathkievua-article-2543 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:28Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cb/bbf84c06d04480844d76dcf378767dcb.pdf |
| spelling | umjimathkievua-article-25432020-03-18T19:26:06Z Lyapunov-Type Inequalities for Quasilinear Systems with Antiperiodic Boundary Conditions Нерівноності типу Ляпунова для квазілінійних систем з антиперіодичними граничними умовами Bai, Yongzhen Li, Yannan Wang, Youyu Бай, Йонгжен Лі, Янань Ван, Й. We establish some new Lyapunov-type inequalities for one-dimensional p-Laplacian systems with antiperiodic boundary conditions. The lower bounds of eigenvalues are presented. Встановлено дєякі нові нєрівності типу Ляпунова для одновимірних p-лапласових систем з антиперіодичними граничними умовами. Наведено нижні межі для власних значень. Institute of Mathematics, NAS of Ukraine 2013-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2543 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 12 (2013); 1646–1656 Український математичний журнал; Том 65 № 12 (2013); 1646–1656 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2543/1846 https://umj.imath.kiev.ua/index.php/umj/article/view/2543/1847 Copyright (c) 2013 Bai Yongzhen; Li Yannan; Wang Youyu |
| spellingShingle | Bai, Yongzhen Li, Yannan Wang, Youyu Бай, Йонгжен Лі, Янань Ван, Й. Lyapunov-Type Inequalities for Quasilinear Systems with Antiperiodic Boundary Conditions |
| title | Lyapunov-Type Inequalities for Quasilinear Systems with Antiperiodic Boundary Conditions |
| title_alt | Нерівноності типу Ляпунова для квазілінійних систем з антиперіодичними граничними умовами |
| title_full | Lyapunov-Type Inequalities for Quasilinear Systems with Antiperiodic Boundary Conditions |
| title_fullStr | Lyapunov-Type Inequalities for Quasilinear Systems with Antiperiodic Boundary Conditions |
| title_full_unstemmed | Lyapunov-Type Inequalities for Quasilinear Systems with Antiperiodic Boundary Conditions |
| title_short | Lyapunov-Type Inequalities for Quasilinear Systems with Antiperiodic Boundary Conditions |
| title_sort | lyapunov-type inequalities for quasilinear systems with antiperiodic boundary conditions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2543 |
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