Lyapunov-type inequalities for two classes of nonlinear systems with homogeneous Dirichlet boundary conditions
We establish new Lyapunov-type inequalities for two classes of nonlinear systems with homogeneous Dirichlet boundary conditions, which generalize and improve some results known from the literature.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507399651065856 |
|---|---|
| author | Aktaş, M. F. Актас, М. Ф. |
| author_facet | Aktaş, M. F. Актас, М. Ф. |
| author_sort | Aktaş, M. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:19:59Z |
| description | We establish new Lyapunov-type inequalities for two classes of nonlinear systems with homogeneous Dirichlet boundary
conditions, which generalize and improve some results known from the literature.
|
| first_indexed | 2026-03-24T02:08:42Z |
| format | Article |
| fulltext |
UDC 517.9
M. F. Aktaş (Gazi Univ., Ankara, Turkey)
LYAPUNOV-TYPE INEQUALITIES FOR TWO CLASSES OF NONLINEAR
SYSTEMS WITH HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS
НЕРIВНОСТI ТИПУ ЛЯПУНОВА ДЛЯ ДВОХ КЛАСIВ НЕЛIНIЙНИХ
СИСТЕМ З ОДНОРIДНИМИ ГРАНИЧНИМИ УМОВАМИ ДIРIХЛЕ
We establish new Lyapunov-type inequalities for two classes of nonlinear systems with homogeneous Dirichlet boundary
conditions, which generalize and improve some results known from the literature.
Встановлено новi нерiвностi типу Ляпунова для двох класiв нелiнiйних систем з однорiдними граничними умовами
Дiрiхле, що узагальнюють та покращують деякi вiдомi результати.
1. Introduction. This paper is concerned with the problem of finding new Lyapunov-type inequali-
ties for the cycled system \bigl(
ri(x)\phi pi
\bigl(
u\prime i
\bigr) \bigr) \prime
+ fi(x)\phi \alpha i (ui+1) = 0 (1.1)
and strongly coupled system
\bigl(
r1(x)\phi p1
\bigl(
u\prime i
\bigr) \bigr) \prime
+ fi(x)
n\sum
j=1
\phi p1 (uj) = 0 (1.2)
for i = 1, 2, . . . , n, where n \in \BbbN , un+1 (x) := u1 (x) and \phi \gamma (u) = | u| \gamma - 2 u with \gamma > 1 under the
following hypotheses:
(H1) rj(x) > 0 and fi(x) for i, j = 1, 2, . . . , n are real-valued continuous functions defined
on \BbbR ,
(H2) the parameters 1 < \alpha i, pi < \infty for i = 1, 2, . . . , n satisfy
\prod n
i=1
\alpha i - 1
pi - 1
= 1.
The well-known Lyapunov inequality [16] for the second-order linear differential equation
u\prime \prime 1 + f1 (x)u1 = 0, (1.3)
states that if f1 (x) is continuous on [a, b] and (1.3) has a real nontrivial solution u1(x) satisfying
the Dirichlet boundary condition u1(a) = 0 = u1(b), then the inequality
4
b - a
\leq
b\int
a
| f1(s)| ds (1.4)
holds, and the constant 4 can not be replaced by a larger number. Since this result has proved to be a
useful tool in oscillation theory, disconjugacy, eigenvalue problems, and many other applications in
c\bigcirc M. F. AKTAŞ, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 6 727
728 M. F. AKTAŞ
the study of various properties of solutions for differential and difference equations, there have been
many proofs and generalizations or improvements of it in the literature. For some of the most recent
works on Lyapunov-type inequalities, the reader is referred to [1 – 35].
Recently, Sim and Lee [23] obtained the following Lyapunov-type inequalities for systems (1.1)
with \alpha i = pi = p1 , ri (x) = 1, and fi (x) \geq 0 for i = 1, 2, . . . , n and (1.2) with r1 (x) = 1 and
fi (x) \geq 0 for i = 1, 2, . . . , n. Their results are as follows:
Theorem A. Let fi \in C ([a, b] , [0,\infty )) for i = 1, 2, . . . , n hold. If system (1.1) with \alpha i =
= pi = p1 and ri (x) = 1 for i = 1, 2, . . . , n has a real nontrivial solution (u1(x), u2(x), . . .
. . . , un (x)) such that ui(a) = 0 = ui(b) for i = 1, 2, . . . , n, where a, b \in \BbbR with a < b are
consecutive zeros and ui for i = 1, 2, . . . , n are not identically zero on [a, b], then the inequality
1 \leq
n\prod
i=1
b\int
a
fi (s)
\Biggl[
2p1 - 2
\biggl(
(s - a) (b - s)
b - a
\biggr) p1 - 1
\Biggr]
ds (1.5)
holds.
Theorem B. Let fi \in C ([a, b] , [0,\infty )) for i = 1, 2, . . . , n hold. If system (1.2) with r1 (x) = 1
has a real nontrivial solution (u1(x), u2(x), . . . , un (x)) such that ui(a) = 0 = ui(b) for i =
= 1, 2, . . . , n, where a, b \in \BbbR with a < b are consecutive zeros and ui for i = 1, 2, . . . , n are
not identically zero on [a, b], then the inequality
1 \leq n
n\sum
i=1
b\int
a
fi (s)
\Biggl[
2p1 - 2
\biggl(
(s - a) (b - s)
b - a
\biggr) p1 - 1
\Biggr]
ds (1.6)
holds.
More recently, Rodrigues [22] generalizes and improves Theorems A and B, respectively, as
follows:
Theorem C. Let fi \in C ([a, b] , [0,\infty )) for i = 1, 2, . . . , n hold. If system (1.1) with \alpha i =
= pi = p1 and ri (x) \geq ki > 0 for i = 1, 2, . . . , n has a real nontrivial solution (u1(x), u2(x), . . .
. . . , un (x)) such that ui(a) = 0 = ui(b) for i = 1, 2, . . . , n, where a, b \in \BbbR with a < b are
consecutive zeros and ui for i = 1, 2, . . . , n are not identically zero on [a, b], then the inequality
1 \leq
\Biggl(
n\prod
i=1
k2i
\Biggr) - 1 n\prod
i=1
b\int
a
fi (s)
\Biggl[
2p1 - 2
\biggl(
(s - a) (b - s)
b - a
\biggr) p1 - 1
\Biggr]
ds (1.7)
holds.
Theorem D. Let fi \in C ([a, b] , [0,\infty )) for i = 1, 2, . . . , n hold. If system
\bigl(
ri(x)\phi p1
\bigl(
u\prime i
\bigr) \bigr) \prime
+ fi(x)
n\sum
j=1
\phi p1 (uj) = 0, (1.8)
where ri (x) \geq ki > 0 for i = 1, 2, . . . , n, has a real nontrivial solution (u1(x), u2(x), . . . , un (x))
such that ui(a) = 0 = ui(b) for i = 1, 2, . . . , n, where a, b \in \BbbR with a < b are consecutive zeros
and ui for i = 1, 2, . . . , n are not identically zero on [a, b], then the inequality
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LYAPUNOV-TYPE INEQUALITIES FOR TWO CLASSES OF NONLINEAR SYSTEMS . . . 729
1 \leq
\bigl(
M + (n - 1)M2
\bigr) n\sum
i=1
b\int
a
fi (s)
\Biggl[
2p1 - 2
\biggl(
(s - a) (b - s)
b - a
\biggr) p1 - 1
\Biggr]
ds (1.9)
holds, where M = \mathrm{m}\mathrm{a}\mathrm{x}i=1,...,n
\biggl\{
1
ki
\biggr\}
.
It is easy to see that if we take M \not = 1, then the inequality (1.9) is better than (1.6) when
M + (n - 1)M2 < n in the sense that (1.6) follows from (1.9), but not conversely. If M = 1, then
Theorem D with ri (x) = 1 for i = 1, 2, . . . , n coincides with Theorem B. Moreover, if we take
1 <
\prod n
i=1
k2i , then the inequality (1.7) is better than (1.5) in the sense that (1.5) follows from (1.7),
but not conversely. Similarly, if
\prod n
i=1
k2i < 1, then Theorem A gives a better result than Theorem
C with ri (x) = 1 for i = 1, 2, . . . , n in the sense that (1.7) follows from (1.5), but not conversely.
If
\prod n
i=1
k2i = 1, then Theorem C with ri (x) = 1 for i = 1, 2, . . . , n coincides with Theorem A.
Throughout the paper, for the sake of brevity, we denote
Di (x) =
1
\xi 1 - pi
i (x) + \eta 1 - pi
i (x)
, Ei (x) = 2pi - 2
\biggl(
1
\xi i(x)
+
1
\eta i(x)
\biggr) 1 - pi
and
Fi = 2 - pi (\xi i (x) + \eta i (x))
pi - 1 = 2 - pi
\left( b\int
a
r
1/(1 - pi)
i (s)ds
\right) pi - 1
,
where
\xi i(x) =
x\int
a
r
1/(1 - pi)
i (s)ds, \eta i(x) =
b\int
x
r
1/(1 - pi)
i (s)ds
for i = 1, 2, . . . , n.
Now, we give some properties of concave and convex functions which are useful in the com-
parison of our results. We know that since the function h(x) = xpi - 1 is concave for x > 0 and
1 < pi < 2, Jensen’s inequality h
\biggl(
\omega + v
2
\biggr)
\geq 1
2
(h(\omega ) + h(v)) with \omega =
1
\xi i(x)
and v =
1
\eta i(x)
implies
Di (x) \geq Ei (x) (1.10)
for 1 < pi < 2, i = 1, 2, . . . , n. If pi > 2, then h(x) = xpi - 1 is convex for x > 0. So, the
inequality (1.10) is reversed, i.e.,
Di (x) \leq Ei (x) (1.11)
for pi > 2, i = 1, 2, . . . , n. In addition, since the function l(x) = x1 - pi is convex for x > 0 and
pi > 1 for i = 1, 2, . . . , n, Jensen’s inequality l
\biggl(
\omega + v
2
\biggr)
\leq 1
2
(l(\omega ) + l(v)) with \omega = \xi i(x) and
v = \eta i(x) implies
Di (x) \leq Fi. (1.12)
Moreover, by using the inequality
4AB \leq (A+B)2
with A = \xi i (x) > 0 and B = \eta i (x) > 0 for i = 1, 2, . . . , n, we obtain the inequality
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 6
730 M. F. AKTAŞ
Ei (x) \leq Fi. (1.13)
In this paper, we state and prove several new generalized Lyapunov-type inequalities for systems
(1.1) and (1.2). Our motivation comes from the recent papers of Rodrigues [22], Sim and Lee [23]
and Tang and He [24]. Our aim is to remove some restrictions on the functions fi (x) and ri (x) for
i = 1, 2, . . . , n in [22]. In fact, we generalize and improve some known results in the literature.
Since our attention is restricted to the Lyapunov-type inequalities for the systems of differential
equations, we shall assume the existence of the nontrivial solution of the system (1.1) or (1.2). For
readers who contributed to the existence of the solution of these type systems, we refer to the paper
by Lee et al. [14].
2. Lyapunov-type inequalities for system (1.1). One of the main results of this section is the
following theorem.
Theorem 2.1. Let the hypotheses (H1) and (H2) hold. If system (1.1) has a real nontrivial
solution (u1(x), u2(x), . . . , un (x)) such that ui(a) = 0 = ui(b) for i = 1, 2, . . . , n, where a,
b \in \BbbR with a < b are consecutive zeros and ui for i = 1, 2, . . . , n are not identically zero on
[a, b], then the inequality
1 <
n\prod
i=1
\left( b\int
a
| fi (s)| D
1
pi
i (s)D
\alpha i - 1
pi+1
i+1 (s) ds
\right)
1
\alpha i - 1
(2.1)
holds, where Dn+1 (x) := D1 (x).
Proof. Let ui(a) = 0 = ui(b), where a, b \in \BbbR with a < b are consecutive zeros and ui for
i = 1, 2, . . . , n are not identically zero on [a, b]. It follows from ui(a) = 0 = ui(b) and the Hölder’s
inequality that
| ui (x)| pi \leq
\left( x\int
a
r
1/(1 - pi)
i (s)ds
\right) pi - 1 x\int
a
ri (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| pi ds = \xi pi - 1
i (x)
x\int
a
ri (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| pi ds (2.2)
and
| ui (x)| pi \leq
\left( b\int
x
r
1/(1 - pi)
i (s)ds
\right) pi - 1 b\int
x
ri (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| pi ds = \eta pi - 1
i (x)
b\int
x
ri (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| pi ds (2.3)
for x \in [a, b] and i = 1, 2, . . . , n. Adding the inequalities (2.2) and (2.3), we have
| ui (x)| pi \leq RiDi (x) , x \in [a, b] , (2.4)
where Ri =
\int b
a
ri (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| pi ds for i = 1, 2, . . . , n. After that by using similar technique to the
proof of Theorem 3.1 in Tang and He [24], it can be showed that the equality case in (2.4) does not
hold. Thus, we can write
| ui (x)| pi < RiDi (x) (2.5)
for x \in (a, b) and i = 1, 2, . . . , n. In addition, we can rewrite the inequality (2.5) as follows:
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 6
LYAPUNOV-TYPE INEQUALITIES FOR TWO CLASSES OF NONLINEAR SYSTEMS . . . 731
| ui+1 (x)| pi+1 < Ri+1Di+1 (x) (2.6)
for x \in (a, b) and i = 0, 1, 2, . . . , n - 1. If we take the 1/pi and (1/pi+1)th powers of both side of
inequalities (2.5) and (2.6), we obtain
| ui (x)| < R
1/pi
i D
1/pi
i (x) (2.7)
for i = 1, 2, . . . , n and
| ui+1 (x)| < R
1/pi+1
i+1 D
1/pi+1
i+1 (x) (2.8)
for i = 0, 1, 2, . . . , n - 1, respectively. On the other hand, for i = 1, 2, . . . , n - 1, multiplying the
ith equation of system (1.1) by ui , integrating from a to b and taking into account that ui(a) =
= 0 = ui(b), we have from the inequalities (2.7) and (2.8)
Ri \leq
b\int
a
| fi (s)| | ui (s)| | ui+1 (s)| \alpha i - 1 ds <
< R
1/pi
i R
(\alpha i - 1)/pi+1
i+1
b\int
a
| fi (s)| D1/pi
i (s)D
(\alpha i - 1)/pi+1
i+1 (s) ds
and hence
R
(pi - 1)/pi
i < R
(\alpha i - 1)/pi+1
i+1
b\int
a
| fi (s)| D1/pi
i (s)D
(\alpha i - 1)/pi+1
i+1 (s) ds (2.9)
for i = 1, 2, . . . , n - 1. Now, for i = n, multiplying the nth equation of system (1.1) by un ,
integrating from a to b and taking into account that un(a) = 0 = un(b), we have from un+1 (x) =
= u1 (x) and the inequality (2.7) with i = 1 and i = n
Rn \leq
b\int
a
| fn (s)| | un (s)| | un+1 (s)| \alpha n - 1 ds =
b\int
a
| fn (s)| | un (s)| | u1 (s)| \alpha n - 1 ds <
< R1/pn
n R
(\alpha n - 1)/p1
1
b\int
a
| fn (s)| D1/pn
n (s)D
(\alpha n - 1)/p1
1 (s) ds
and hence
R(pn - 1)/pn
n < R
(\alpha n - 1)/p1
1
b\int
a
| fn (s)| D1/pn
n (s)D
(\alpha n - 1)/p1
1 (s) ds (2.10)
for n \in \BbbN . Raising the both sides of inequalities (2.9) and (2.10), to the power ei for each i =
= 1, 2, . . . , n - 1 and en , respectively, and multiplying the resulting inequalities side by side, we
obtain
n\prod
i=1
R
pi - 1
pi
ei
i <
n\prod
i=1
R
\alpha i - 1
pi+1
ei
i+1
n\prod
i=1
\left( b\int
a
| fi (s)| D
1
pi
i (s)D
\alpha i - 1
pi+1
i+1 (s) ds
\right) ei
and hence
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 6
732 M. F. AKTAŞ
R
p1 - 1
p1
e1 - \alpha n - 1
p1
en
1
n\prod
i=2
R
pi - 1
pi
ei -
\alpha i - 1 - 1
pi
ei - 1
i <
n\prod
i=1
\left( b\int
a
| fi (s)| D
1
pi
i (s)D
\alpha i - 1
pi+1
i+1 (s) ds
\right) ei
. (2.11)
Next, we prove that Ri > 0 for i = 1, 2, . . . , n. If the inequality Ri > 0 is not true, then Ri = 0
for i = 1, 2, . . . , n. If Ri =
\int b
a
ri (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| pi ds = 0, then it follows that
u\prime i (x) \equiv 0 (2.12)
for a \leq x \leq b and i = 1, 2, . . . , n. Combining (2.2) with (2.12), we obtain that ui(x) \equiv 0 for
a \leq x \leq b, which contradicts ui(x) \not \equiv 0 for a \leq x \leq b and i = 1, 2, . . . , n. Therefore, Ri > 0
for i = 1, 2, . . . , n holds. Now, we find a relation between \alpha i and pi for i = 1, 2, . . . , n such that
Ri > 0 for i = 1, 2, . . . , n cancel out in the inequality (2.11), i.e., solve the homogeneous linear
system
(p1 - 1) e1 - (\alpha n - 1) en = 0,
(pi - 1) ei - (\alpha i - 1 - 1) ei - 1 = 0
for i = 2, 3, . . . , n. We observe that by hypothesis
\prod n
i=1
\alpha i - 1
pi - 1
= 1, this system admits a nontrivial
solution. Hence, we may take ei = e1
p1 - 1
\alpha i - 1
where e1 > 0 and i = 1, 2, . . . , n. Therefore, we
obtain the inequality (2.1).
Theorem 2.1 is proved.
Another main result of this section is the following theorem.
Theorem 2.2. Let the hypotheses (H1) and (H2) hold. If system (1.1) has a real nontrivial
solution (u1(x), u2(x), . . . , un (x)) such that ui(a) = 0 = ui(b) for i = 1, 2, . . . , n, where a, b \in \BbbR
with a < b are consecutive zeros and ui for i = 1, 2, . . . , n are not identically zero on [a, b], then
the inequality
1 <
n\prod
i=1
\left( b\int
a
| fi (s)| E
1
pi
i (s)E
\alpha i - 1
pi+1
i+1 (s) ds
\right)
1
\alpha i - 1
(2.13)
holds, where En+1 (x) := E1 (x).
Proof. Let ui(a) = 0 = ui(b), where a, b \in \BbbR with a < b are consecutive zeros and ui
for i = 1, 2, . . . , n are not identically zero on [a, b]. As in the proof of Theorem 2.1, we have the
inequalities (2.2) and (2.3). Multiplying the inequalities (2.2) and (2.3) by \eta pi - 1
i (x) and \xi pi - 1
i (x),
i = 1, 2, . . . , n, respectively, we obtain
\eta pi - 1
i (x) | ui (x)| pi \leq (\xi i(x)\eta i(x))
pi - 1
x\int
a
ri (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| pi ds (2.14)
and
\xi pi - 1
i (x) | ui (x)| pi \leq (\xi i(x)\eta i(x))
pi - 1
b\int
x
ri (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| pi ds (2.15)
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LYAPUNOV-TYPE INEQUALITIES FOR TWO CLASSES OF NONLINEAR SYSTEMS . . . 733
for x \in [a, b] and i = 1, 2, . . . , n. Thus, adding the inequalities (2.14) and (2.15), we have
| ui (x)| pi
\Bigl(
\xi pi - 1
i (x) + \eta pi - 1
i (x)
\Bigr)
\leq (\xi i(x)\eta i(x))
pi - 1
b\int
a
ri (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| pi ds
for x \in [a, b] and i = 1, 2, . . . , n. It is easy to see that the functions \xi pi - 1
i (x) + \eta pi - 1
i (x) take the
minimum values at ci \in (a, b) such that \xi i (ci) = \eta i (ci) for i = 1, 2, . . . , n. Thus, we get
| ui (x)| pi
\Bigl(
\xi pi - 1
i (ci) + \eta pi - 1
i (ci)
\Bigr)
\leq (\xi i(x)\eta i(x))
pi - 1
b\int
a
ri (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| pi ds
for i = 1, 2, . . . , n. Since \xi i(ci)+\eta i(ci) = \xi i(x)+\eta i(x) \forall x, ci \in (a, b), and \xi i(ci) =
\xi i(x) + \eta i(x)
2
=
=
1
2
\int b
a
r
1/(1 - pi)
i (s)ds, we obtain
| ui (x)| pi
\Bigl[
22 - pi (\xi i(x) + \eta i(x))
pi - 1
\Bigr]
= | ui (x)| pi
\Bigl[
2\xi pi - 1
i (ci)
\Bigr]
\leq
\leq (\xi i(x)\eta i(x))
pi - 1
b\int
a
ri (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| pi ds
and hence
| ui (x)| pi \leq RiEi (x) , (2.16)
where Ri =
\int b
a
ri (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| pi ds for i = 1, 2, . . . , n. After that by using similar technique to the
proof of Theorem 3.1 in Tang and He [24], it can be showed that the equality case in (2.16) does not
hold. Thus, we can write
| ui (x)| pi < RiEi (x)
for x \in (a, b) and i = 1, 2, . . . , n. The rest of the proof is the same as in the proof of Theorem 2.1,
and hence is omitted.
Theorem 2.2 is proved.
Remark 2.1. Note that pi > 1 for i = 1, 2, . . . , n in both Theorems 2.1 and 2.2. However,
it is easy to see from the inequality (1.10) that if we take 1 < pi < 2 for i = 1, 2, . . . , n, then the
inequality (2.13) is better than (2.1) in the sense that (2.1) follows from (2.13), but not conversely.
Similarly, from the inequality (1.11), if pi > 2 for i = 1, 2, . . . , n, then the inequality (2.1) is better
than (2.13) in the sense that (2.13) follows from (2.1), but not conversely. In addition, if pi = 2 for
i = 1, 2, . . . , n, then Theorem 2.1 coincides with Theorem 2.2.
By using the inequality (1.12) in Theorem 2.1 or (1.13) in Theorem 2.2, we obtain the following
result.
Corollary 2.1. Let the hypotheses (H1) and (H2) hold. If system (1.1) has a real nontrivial
solution (u1(x), u2(x), . . . , un (x)) such that ui(a) = 0 = ui(b) for i = 1, 2, . . . , n, where a, b \in \BbbR
with a < b are consecutive zeros and ui for i = 1, 2, . . . , n are not identically zero on [a, b], then
the inequality
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 6
734 M. F. AKTAŞ
1 <
n\prod
i=1
\left( F
1
pi
i F
\alpha i - 1
pi+1
i+1
b\int
a
| fi (s)| ds
\right)
1
\alpha i - 1
holds, where Fn+1 := F1 .
If we get the conditions \alpha i = pi = p1 and ri (x) \geq ki > 0 for i = 1, 2, . . . , n in Theorems 2.1
and 2.2, then we obtain the following results, respectively.
Corollary 2.2. Let the hypotheses (H1) and (H2) hold. If system (1.1) with \alpha i = pi = p1 and
ri (x) \geq ki > 0 for i = 1, 2, . . . , n has a real nontrivial solution (u1(x), u2(x), . . . , un (x)) such
that ui(a) = 0 = ui(b) for i = 1, 2, . . . , n, where a, b \in \BbbR with a < b are consecutive zeros and
ui for i = 1, 2, . . . , n are not identically zero on [a, b], then the inequality
1 <
\Biggl(
n\prod
i=1
k
1
p1
i k
1 - 1
p1
i+1
\Biggr) - 1 n\prod
i=1
b\int
a
| fi (s)|
\biggl[
1
(s - a)1 - p1 + (b - s)1 - p1
\biggr]
ds
holds, where kn+1 := k1 .
Corollary 2.3. Let the hypotheses (H1) and (H2) hold. If system (1.1) with \alpha i = pi = p1 and
ri (x) \geq ki > 0 for i = 1, 2, . . . , n has a real nontrivial solution (u1(x), u2(x), . . . , un (x)) such
that ui(a) = 0 = ui(b) for i = 1, 2, . . . , n, where a, b \in \BbbR with a < b are consecutive zeros and
ui for i = 1, 2, . . . , n are not identically zero on [a, b], then the inequality
1 <
\Biggl(
n\prod
i=1
k
1
p1
i k
1 - 1
p1
i+1
\Biggr) - 1 n\prod
i=1
b\int
a
| fi (s)|
\Biggl[
2p1 - 2
\biggl(
b - a
(s - a) (b - s)
\biggr) 1 - p1
\Biggr]
ds (2.17)
holds, where kn+1 := k1 .
Remark 2.2. Note that since kn+1 := k1, we obtain
\prod n
i=1
k
1
p1
i k
1 - 1
p1
i+1 =
\prod n
i=1
ki. Let fi (x) \geq
\geq 0 for i = 1, 2, . . . , n hold. It is easy to see that if we take 0 <
\prod n
i=1
ki < 1, then the inequality
(2.17) is better than (1.7) in the sense that (1.7) follows from (2.17), but not conversely. Similarly,
if
\prod n
i=1
ki > 1, then the inequality (1.7) is better than (2.17) in the sense that (2.17) follows from
(1.7), but not conversely. In addition, if
\prod n
i=1
ki = 1, then Corollary 2.2 coincides with Theorem C
given by Rodrigues [22].
Remark 2.3. Let \alpha i = pi = p1, ri (x) = 1, and fi (x) \geq 0 for i = 1, 2, . . . , n hold. It is easy
to see from the inequality (1.10) that if we take 1 < p1 < 2, then Theorem A given by Sim and
Lee [23] is better than Theorem 2.1 in the sense that (2.1) follows from (1.5), but not conversely.
Similarly, from the inequality (1.11), if p1 > 2, then Theorem 2.1 is better than Theorem A in the
sense that (1.5) follows from (2.1), but not conversely. In addition, if p1 = 2, then Theorems A, 2.1,
and 2.2 are equivalent to each other.
3. Lyapunov-type inequalities for system (1.2). For system (1.2), one of the main results of
this section is the following theorem.
Theorem 3.1. Let the hypothesis (H1) with j = 1 hold. If system (1.2) has a real nontrivial
solution (u1(x), u2(x), . . . , un (x)) such that ui(a) = 0 = ui(b) for i = 1, 2, . . . , n, where a,
b \in \BbbR with a < b are consecutive zeros and ui for i = 1, 2, . . . , n are not identically zero on [a, b],
then the inequality
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LYAPUNOV-TYPE INEQUALITIES FOR TWO CLASSES OF NONLINEAR SYSTEMS . . . 735
1 <
n\sum
i=1
b\int
a
| fi (s)| D1 (s) ds (3.1)
holds.
Proof. Let ui(a) = 0 = ui(b) where a, b \in \BbbR with a < b are consecutive zeros, and ui for
i = 1, 2, . . . , n are not identically zero on [a, b]. It follows from the similar technique to the proof of
Theorem 2.1, we obtain the inequalities
| ui (x)| p1 < MiD1 (x) , (3.2)
| ui (x)| < M
1/p1
i D
1/p1
1 (x) , (3.3)
and
| ui (x)| p1 - 1 < M
(p1 - 1)/p1
i D
(p1 - 1)/p1
1 (x) , (3.4)
where Mi =
\int b
a
r1 (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| p1 ds for i = 1, 2, . . . , n. On the other hand, multiplying the ith
equation of system (1.2) by ui , integrating from a to b and taking into account that ui(a) = 0 =
= ui(b), we have from the inequalities (3.2), (3.3), and (3.4) that
Mi \leq
b\int
a
| fi (s)| | ui (s)| p1 ds+
b\int
a
| fi (s)| | ui (s)|
n\sum
j=1
j \not =i
| uj (s)| p1 - 1 ds <
< Mi
b\int
a
| fi (s)| D1 (s) ds+M
1/p1
i
n\sum
j=1
j \not =i
M
(p1 - 1)/p1
j
b\int
a
| fi (s)| D1 (s) ds
and hence
M
(p1 - 1)/p1
i <
n\sum
j=1
M
(p1 - 1)/p1
j
b\int
a
| fi (s)| D1 (s) ds (3.5)
for i = 1, 2, . . . , n. It is easy to see that by using similar technique to the proof of Theorem 2.1, we
obtain Mi > 0 for i = 1, 2, . . . , n. By summing the inequalities (3.5), we have
n\sum
i=1
M
(p1 - 1)/p1
i <
n\sum
j=1
M
(p1 - 1)/p1
j
n\sum
i=1
b\int
a
| fi (s)| D1 (s) ds
and hence
1 <
n\sum
i=1
b\int
a
| fi (s)| D1 (s) ds,
which completes the proof.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 6
736 M. F. AKTAŞ
Theorem 3.2. Let the hypothesis (H1) with j = 1 hold. If system (1.2) has a real nontrivial
solution (u1(x), u2(x), . . . , un (x)) such that ui(a) = 0 = ui(b) for i = 1, 2, . . . , n, where a,
b \in \BbbR with a < b are consecutive zeros and ui for i = 1, 2, . . . , n are not identically zero on [a, b],
then the inequality
1 <
n\sum
i=1
b\int
a
| fi (s)| E1 (s) ds (3.6)
holds.
Proof. Let ui(a) = 0 = ui(b), where a, b \in \BbbR with a < b are consecutive zeros and ui for
i = 1, 2, . . . , n are not identically zero on [a, b]. It follows from the similar technique to the proof of
Theorem 2.2, we have the inequalities
| ui (x)| p1 < MiE1 (x) ,
| ui (x)| < M
1/p1
i E
1/p1
1 (x) ,
and
| ui (x)| p1 - 1 < M
(p1 - 1)/p1
i E
(p1 - 1)/p1
1 (x) ,
where Mi =
\int b
a
r1 (s)
\bigm| \bigm| u\prime i (s)\bigm| \bigm| p1 ds for i = 1, 2, . . . , n. The rest of the proof is the same as in the
proof of Theorem 3.1 and hence is omitted.
Remark 3.1. Note that p1 > 1 in both Theorems 3.1 and 3.2. However, it is easy to see from
the inequality (1.10) that if we take 1 < p1 < 2, then the inequality (3.6) is better than (3.1) in
the sense that (3.1) follows from (3.6), but not conversely. Similarly, from the inequality (1.11), if
p1 > 2, then the inequality (3.1) is better than (3.6) in the sense that (3.6) follows from (3.1), but not
conversely. In addition, if p1 = 2, then Theorem 3.1 coincides with Theorem 3.2.
By using the inequality (1.12) in Theorem 3.1 or (1.13) in Theorem 3.2, we have the following
result.
Corollary 3.1. Let the hypothesis (H1) with j = 1 hold. If system (1.2) has a real nontrivial
solution (u1(x), u2(x), . . . , un (x)) such that ui(a) = 0 = ui(b) for i = 1, 2, . . . , n, where a,
b \in \BbbR with a < b are consecutive zeros and ui for i = 1, 2, . . . , n are not identically zero on
[a, b], then the inequality
1 < F1
n\sum
i=1
b\int
a
| fi (s)| ds
holds.
If we get the condition r1 (x) \geq k1 > 0 in Theorems 3.1 and 3.2, we obtain the following results,
respectively.
Corollary 3.2. Let the hypothesis (H1) with j = 1 hold. If system (1.2) with r1 (x) \geq k1 > 0
has a real nontrivial solution (u1(x), u2(x), . . . , un (x)) such that ui(a) = 0 = ui(b) for i =
= 1, 2, . . . , n, where a, b \in \BbbR with a < b are consecutive zeros and ui for i = 1, 2, . . . , n are
not identically zero on [a, b], then the inequality
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LYAPUNOV-TYPE INEQUALITIES FOR TWO CLASSES OF NONLINEAR SYSTEMS . . . 737
1 <
1
k1
n\sum
i=1
b\int
a
| fi (s)|
\biggl[
1
(s - a)1 - p1 + (b - s)1 - p1
\biggr]
ds (3.7)
holds.
Corollary 3.3. Let the hypothesis (H1) with j = 1 hold. If system (1.2) with r1 (x) \geq k1 > 0
has a real nontrivial solution (u1(x), u2(x), . . . , un (x)) such that ui(a) = 0 = ui(b) for i =
= 1, 2, . . . , n, where a, b \in \BbbR with a < b are consecutive zeros and ui for i = 1, 2, . . . , n are
not identically zero on [a, b], then the inequality
1 <
1
k1
n\sum
i=1
b\int
a
| fi (s)|
\Biggl[
2p1 - 2
\biggl(
(s - a) (b - s)
b - a
\biggr) p1 - 1
\Biggr]
ds (3.8)
holds.
Remark 3.2. Let r1 (x) \geq k1 > 0 in system (1.2), ri (x) = r1 (x) \geq k1 > 0 for i = 1, 2, . . . , n
in system (1.8), and fi (x) \geq 0 for i = 1, 2, . . . , n hold. It is easy to see from the inequality (1.11)
that if we take p1 > 2, then the inequality (3.7) is better than (1.9) in the sense that (1.9) follows
from (3.7), but not conversely. Moreover, it is easy to see that the inequality (3.8) is better than (1.9)
in the sense that (1.9) follows from (3.8), but not conversely. In addition, if n = 1, then Corollary
3.2 coincides with Theorem D given by Rodrigues [22].
Remark 3.3. Let fi (x) \geq 0 for i = 1, 2, . . . , n and r1 (x) = 1. If we take n \not = 1, the inequality
(3.6) is better than (1.6) in the sense that (1.6) follows from (3.6), but not conversely. Thus, Theorem
3.2 improves and generalizes Theorem B given by Sim and Lee [23]. If n = 1, then Theorem 3.2
coincides with Theorem B. Moreover, from the inequality (1.11), if p1 > 2, then Theorem 3.1 gives
a better result than Theorem B. If n = 1 and p1 = 2, then Theorems 3.1, 3.2, and B are equivalent
to each other.
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Received 21.10.14
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|
| id | umjimathkievua-article-1591 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:08:42Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/75/7e3dec868993a76d68e5a872cb99d675.pdf |
| spelling | umjimathkievua-article-15912019-12-05T09:19:59Z Lyapunov-type inequalities for two classes of nonlinear systems with homogeneous Dirichlet boundary conditions Нерiвностi типу Ляпунова для двох класiв нелiнiйних систем з однорiдними граничними умовами Дiрiхле Aktaş, M. F. Актас, М. Ф. We establish new Lyapunov-type inequalities for two classes of nonlinear systems with homogeneous Dirichlet boundary conditions, which generalize and improve some results known from the literature. Встановлено новi нерiвностi типу Ляпунова для двох класiв нелiнiйних систем з однорiдними граничними умовами Дiрiхле, що узагальнюють та покращують деякi вiдомi результати. Institute of Mathematics, NAS of Ukraine 2018-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1591 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 6 (2018); 727-738 Український математичний журнал; Том 70 № 6 (2018); 727-738 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1591/573 Copyright (c) 2018 Aktaş M. F. |
| spellingShingle | Aktaş, M. F. Актас, М. Ф. Lyapunov-type inequalities for two classes of nonlinear systems with homogeneous Dirichlet boundary conditions |
| title | Lyapunov-type inequalities for two classes of nonlinear systems with
homogeneous Dirichlet boundary conditions |
| title_alt | Нерiвностi типу Ляпунова для двох класiв нелiнiйних
систем з однорiдними граничними умовами Дiрiхле |
| title_full | Lyapunov-type inequalities for two classes of nonlinear systems with
homogeneous Dirichlet boundary conditions |
| title_fullStr | Lyapunov-type inequalities for two classes of nonlinear systems with
homogeneous Dirichlet boundary conditions |
| title_full_unstemmed | Lyapunov-type inequalities for two classes of nonlinear systems with
homogeneous Dirichlet boundary conditions |
| title_short | Lyapunov-type inequalities for two classes of nonlinear systems with
homogeneous Dirichlet boundary conditions |
| title_sort | lyapunov-type inequalities for two classes of nonlinear systems with
homogeneous dirichlet boundary conditions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1591 |
| work_keys_str_mv | AT aktasmf lyapunovtypeinequalitiesfortwoclassesofnonlinearsystemswithhomogeneousdirichletboundaryconditions AT aktasmf lyapunovtypeinequalitiesfortwoclassesofnonlinearsystemswithhomogeneousdirichletboundaryconditions AT aktasmf nerivnostitipulâpunovadlâdvohklasivnelinijnihsistemzodnoridnimigraničnimiumovamidirihle AT aktasmf nerivnostitipulâpunovadlâdvohklasivnelinijnihsistemzodnoridnimigraničnimiumovamidirihle |