Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 67, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
LYAPUNOV-TYPE INEQUALITIES FOR n-DIMENSIONAL
QUASILINEAR SYSTEMS
MUSTAFA FAHRI AKTAŞ
Abstract. In this article, inspired by the paper of Yang et al [12], we establish new versions of Lyapunov-type inequalities for a certain class of Dirichlet
quasilinear systems.
1. Introduction
In this article, we prove generalized Lyapunov-type inequalities for a special case
of the system
n
Y
0
− rk (x)φpk (u0k ) = fk (x)φαkk (uk )
|ui |αki ,
(1.1)
i=1, i6=k
γ−2
where n ∈ N, φγ (u) = |u|
u, γ > 1, rk , fk ∈ C([a, b], R), rk (x) > 0 for k =
1, 2, . . . , n and x ∈ R. Let (u1 (x), u2 (x), . . . , un (x)) be a real nontrivial solution of
the system (1.1) such that
uk (a) = uk (b) = 0 k = 1, 2, . . . , n,
(1.2)
for a, b ∈ R with a < b are consecutive zeros of uk , and uk are not identically zero
on [a, b], 1 < pk < ∞ and αki are nonnegative constants, for k, i = 1, 2, . . . , n.
Lyapunov [6] proved the following remarkable result, for problem (1.1)-(1.2) with
n = 1, p1 = 2, and r1 (x) = 1,
−u001 = f1 (x)u1 ,
(1.3)
u1 (a) = u1 (b) = 0.
(1.4)
Theorem 1.1. If f1 ∈ C([a, b], [0, ∞)) and u1 (x) is a nontrivial solution on [a, b]
for the problem (1.3)-(1.4), then the so-called Lyapunov inequality holds,
Z b
4
≤
f1 (s)ds .
(1.5)
b−a
a
Lyapunov-type inequalities have been studied extensively; see for example the
references in this article and their references. Çakmak and Tiryaki [3] obtained the
P2
following inequality for system (1.1) with n = 2 under the condition k=1 αpik
=1
k
for i = 1, 2.
2000 Mathematics Subject Classification. 34A40.
Key words and phrases. Lyapunov-type inequality; quasilinear system.
c
2013
Texas State University - San Marcos.
Submitted January 4, 2013. Published March 5, 2013.
1
2
M. F. AKTAŞ
EJDE-2013/67
Theorem 1.2. If fk ∈ C([a, b], R) for k = 1, 2 and (u1 (x), u2 (x)) is a nontrivial
solution on [a, b] for the system (1.1) with n = 2, then the inequality
Z b
αp21 Z b
αp12
1
2
+
+
α21 +α12
2
≤
f1 (s)ds
f2 (s)ds
a
a
(1.6)
Z b
α21 (p1 −1) Z b
α12 (p2 −1)
1
×
(r1 (s)) 1−p1 ds
1
p1
(r2 (s)) 1−p2 ds
a
holds, where
fk+ (x)
p2
a
= max{0, fk (x)} is the nonnegative part of fk (x) for k = 1, 2.
Recently, Çakmak and Tiryaki [4] obtained the following inequality for system
with rk (x) = 1 and αik = αkk , k, i = 1, 2, . . . , n, under the condition
Pn (1.1)
αkk
k=1 pk = 1.
Theorem 1.3. If fk ∈ C([a, b], R) for and (u1 (x), u2 (x), . . . , un (x)) is a nontrivial
solution on [a, b] for system (1.1) with rk (x) = 1 and αik = αkk , k, i = 1, 2, . . . , n,
then the inequality
n
n Z b
αkk /pk
Y
Y
[(ck − a)1−pk + (b − ck )1−pk ]αkk /pk ≤
fk+ (s)ds
(1.7)
k=1
a
k=1
holds, where |uk (ck )| = maxa<x<b |uk (x)| and fk+ (x) = max{0, fk (x)} for k =
1, 2, . . . , n.
Throughout this article, for the sake of brevity, we denote
ξ (x)η (x) pk −1
[ξk (x)ηk (x)]pk −1
k
k
, Ek (x) = 2pk −2
,
Dk (x) = pk −1
pk −1
ξ
(x)
+
ηk (x)
ξk
(x) + ηk
(x)
k
Z b
pk −1
1/(1−pk )
Fk = 2−pk (ξk (x) + ηk (x))pk −1 = 2−pk
,
rk
(s)ds
(1.8)
(1.9)
a
where
Z
ξk (x) =
a
x
1/(1−pk )
rk
Z
(s)ds,
b
ηk (x) =
x
1/(1−pk )
rk
(s)ds
(1.10)
for k = 1, 2, . . . , n.
Recently, Yang et al. [12] obtained the following inequality for system (1.1).
Theorem 1.4. Assume that there exist nontrivial solutions (e1 , e2 , . . . , en ) of the
linear homogeneous system
ek 1 −
αkk −
pk
n
X
i=1, i6=k
αik
ei = 0,
pk
(1.11)
Pn
where ek ≥ 0 for k = 1, 2, . . . , n and k=1 e2k > 0. If fk ∈ C([a, b], R) for k =
1, 2, . . . , n and (u1 (x), u2 (x), . . . , un (x)) is a nontrivial solution on [a, b] for system
(1.1), then the inequality
Z b
n ek
Y
(1.12)
1<
Fk
fk+ (s)ds
k=1
a
holds, where fk+ (x) = max{0, fk (x)} for k = 1, 2, . . . n.
EJDE-2013/67
LYAPUNOV-TYPE INEQUALITIES
3
Our motivation comes from the recent papers of Çakmak and Tiryaki [3, 4],
Sim and Lee [8], Tang and He [9], and Yang et al. [12]. In this article, we state
and prove new generalized Lyapunov-type inequalities for system (1.1) under the
condition αki = αik for k, i = 1, 2, . . . , n.
Since our attention is restricted to the Lyapunov-type inequalities for the quasilinear systems of differential equations, we shall assume the existence of the nontrivial solution of the system (1.1). For readers interested in the existence of solutions
of these type systems, we refer to the paper by Afrouzi and Heidarkhani [1].
Now, we present some inequalities on Dk (x), Ek (x), and Fk for k = 1, 2, . . . , n
which are useful in the comparison of our main results. We know that since the
function h(x) = xpk −1 is concave for x > 0 and 1 < pk < 2, Jensen’s inequality
1
h( ω+v
2 ) ≥ 2 [h(ω) + h(v)] with ω = 1/ξk (x) and v = 1/ηk (x) implies
Dk (x) ≥ Ek (x)
(1.13)
for 1 < pk < 2, k = 1, 2, . . . , n. If pk > 2 for k = 1, 2, . . . , n, then the function
h(x) = xpk −1 is convex for x > 0. Thus, the inequality (1.13) is reversed; i.e.,
Dk (x) ≤ Ek (x)
(1.14)
for pk > 2, k = 1, 2, . . . , n. In addition, since the function l(x) = x1−pk is convex
1
for x > 0 and pk > 1, Jensen’s inequality l( ω+v
2 ) ≤ 2 [l(ω) + l(v)] with ω = ξk (x)
and v = ηk (x) implies
Dk (x) ≤ Fk
(1.15)
for k = 1, 2, . . . , n. By using inequality
4AB ≤ (A + B)2
(1.16)
with A = ξk (x) > 0 and B = ηk (x) > 0 for k = 1, 2, . . . , n in Ek (x), we obtain the
inequality
Ek (x) ≤ Fk
(1.17)
for k = 1, 2, . . . , n.
2. Main results
One of the main results of this paper is the following theorem.
Theorem 2.1. Assume that there exist nontrivial solutions (e1 , e2 , . . . , en ) of the
linear homogeneous system
ek
αkk 1−
−
pk
n
X
i=1, i6=k
αki
ei = 0,
pk
(2.1)
Pn
where ek ≥ 0 for k = 1, 2, . . . , n and k=1 e2k > 0. If fk ∈ C([a, b], R) for k =
1, 2, . . . , n and (u1 (x), u2 (x), . . . , un (x)) is a nontrivial solution on [a, b] for system
(1.1) with αki = αik for k, i = 1, 2, . . . , n, then the inequality
n hZ b
n
i ek
Y
Y
α /p
1<
fk+ (s)
Di ki i (s)ds
(2.2)
k=1
holds, where
fk+ (x)
a
i=1
= max{0, fk (x)} for k = 1, 2, . . . n.
4
M. F. AKTAŞ
EJDE-2013/67
Proof. Let uk (a) = 0 = uk (b) for k = 1, 2, . . . , n where n ∈ N, a, b ∈ R with a < b
are consecutive zeros and uk for k = 1, 2, . . . , n are not identically zero on [a, b]. By
using uk (a) = 0 and Hölder’s inequality, we obtain
Z x
|uk (x)| ≤
|u0k (s)|ds
a
Z x
(pk −1)/pk Z x
1/pk
1/(1−pk )
≤
rk
(s)ds
rk (s)|u0k (s)|pk ds
a
a
Z x
1/pk
(pk −1)/pk
= ξk
(x)
rk (s)|u0k (s)|pk ds
a
for k = 1, 2, . . . , n and x ∈ [a, b]. Thus, we have
Z x
1−pk
|uk (x)|pk ξk (x) ≤
rk (s)|u0k (s)|pk ds
(2.3)
a
for k = 1, 2, . . . , n and x ∈ [a, b]. Similarly, by using uk (b) = 0 and Hölder’s
inequality, we obtain
Z b
pk 1−pk
|uk (x)| ηk (x) ≤
rk (s)|u0k (s)|pk ds
(2.4)
x
for k = 1, 2, . . . , n and x ∈ [a, b]. Adding (2.3) and (2.4), we have
Z b
|uk (x)|pk ≤ Dk (x)
rk (s)|u0k (s)|pk ds
(2.5)
a
for k = 1, 2, . . . , n and x ∈ [a, b]. After that by using a technique similar to the one
in [9, Theorem 3.1], it can be showed that the equality case in (2.5) does not hold.
Thus, we have
|uk (x)|pk < Ak Dk (x), x ∈ (a, b),
(2.6)
Rb
-th power of
where Ak = a rk (s)|u0k (s)|pk ds for k = 1, 2, . . . , n. If we take the αpki
k
both side of inequality (2.6), we obtain
α
|uk (x)|αki < Ak ki
/pk
α
Dk ki
/pk
(x),
(2.7)
for k, i = 1, 2, . . . , n.
Qn
Multiplying both sides of (2.7) with i = k by fk+ (x) i=1, i6=k |ui (x)|αki for k =
1, 2, . . . , n, integrating from a to b, we have
Z b
Z b
n
n
Y
Y
α /p
α /p
fk+ (s)Dk kk k (s)
|ui (s)|αki ds (2.8)
fk (s)
|ui (s)|αki ds < Ak kk k
a
a
i=1
i=1, i6=k
for k = 1, 2, . . . , n. On the other hand, multiplying the k-th equation of system
(1.1) by uk and integrating from a to b, we get
Z b
Z b
n
Y
Ak =
rk (s)|u0k (s)|pk ds =
fk (s)
|ui (s)|αki ds
(2.9)
a
a
i=1
for k = 1, 2, . . . , n. By using (2.9) in (2.8), we have
Z b
n
Y
α /p
α /p
Ak < Ak kk k
fk+ (s)Dk kk k (s)
a
i=1, i6=k
|ui (s)|αki ds
EJDE-2013/67
LYAPUNOV-TYPE INEQUALITIES
and hence from αki = αik for k, i = 1, 2, . . . , n
Z b
α /p
α /p
Ak < Ak kk k
fk+ (s)Dk kk k (s)
a
n
Y
5
|ui (s)|αik ds
(2.10)
i=1, i6=k
for k = 1, 2, . . . , n. Therefore, by using (2.7) in (2.10), we have
Z b
n
Y
1−αkk /pk
αkk /pk
α /p
α /p
+
Ak
<
fk (s)Dk
(s)
Ai ik i Di ik i (s)ds,
a
i=1, i6=k
and hence
1−αkk /pk
Ak
n
Y
<
α
Ai ik
/pi
b
Z
a
i=1, i6=k
fk+ (s)
n
Y
α
Di ik
/pi
(s)ds
(2.11)
i=1
for k = 1, 2, . . . , n. Raising the both sides of the inequality (2.11) to the power ek
for each k = 1, 2, . . . , n, respectively, and multiplying the resulting inequalities side
by side, we obtain
n
n h Y
n
n hZ b
n
i ek Y
i ek
Y
Y
Y
e (1−αkk /pk )
α /p
α /p
Akk
<
Ai ik i
fk+ (s)
Di ik i (s)ds
k=1
k=1
i=1, i6=k
and hence
Pn
n
n
hY
Y
i=1, i6=k
e (1−αkk /pk )
Ak
Akk
<
αki
pk
ei i
a
n Z
Y
[
b
k=1
k=1
k=1
k=1
a
i=1
fk+ (s)
n
Y
α
Di ik
/pi
(s)ds]ek .
i=1
(2.12)
It is easy to see that by using a technique similar to the one in [9, Theorem 3.1],
we obtain the inequalities Ak > 0 for k = 1, 2, . . . , n. Thus, we have
n
n hZ b
n
i ek
Y
Y
Y
α /p
θk
+
Ak <
Di ki i (s)ds ,
(2.13)
fk (s)
k=1
a
k=1
i=1
Pn
αkk
αki
i=1, i6=k pk ei
pk )−
where θk = ek (1−
for k = 1, 2, . . . , n. By assumption, system
(2.1) has nonzero solutions (e1 , e2 , . . . , en ) such that θk = 0 for k = 1, 2, . . . , n,
where ek ≥ 0 for k = 1, 2, . . . , n and at least one ej > 0 for j = {1, 2, . . . , n}.
Choosing one of the solutions (e1 , e2 , . . . en ), we obtain from (2.13) the inequality
(2.2). This completes the proof.
Another main result of this paper is the following theorem.
Theorem 2.2. Assume that there exist nontrivial solutions (e1 , e2 , . . . , en ) of system (2.1). If fk ∈ C([a, b], R) for k = 1, 2, . . . , n and (u1 (x), u2 (x), . . . , un (x)) is a
nontrivial solution on [a, b] for the system (1.1) with αki = αik for k, i = 1, 2, . . . , n,
then the inequality
n hZ b
n
iek
Y
Y
α /p
1<
fk+ (s)
Ei ki i (s)ds
(2.14)
k=1
holds, where
fk+ (x)
a
i=1
= max{0, fk (x)} for k = 1, 2, . . . n.
Proof. Let uk (a) = 0 = uk (b) for k = 1, 2, . . . , n where n ∈ N, a, b ∈ R with a < b
are consecutive zeros and uk for k = 1, 2, . . . , n are not identically zero on [a, b]. As
in the proof of Theorem 2.1, we have inequalities (2.3) and (2.4). Multiplying the
6
M. F. AKTAŞ
EJDE-2013/67
inequalities (2.3) and (2.4) by ηkpk −1 (x) and ξkpk −1 (x), k = 1, 2, . . . , n, respectively,
we obtain
Z x
rk (s)|u0k (s)|pk ds
(2.15)
ηkpk −1 (x)|uk (x)|pk ≤ ηkpk −1 (x)ξkpk −1 (x)
a
and
ξkpk −1 (x)|uk (x)|pk ≤ ξkpk −1 (x)ηkpk −1 (x)
Z
b
rk (s)|u0k (s)|pk ds
(2.16)
x
for k = 1, 2, . . . , n and x ∈ [a, b]. Thus, adding the inequalities (2.15) and (2.16),
we have
Z b
pk −1
pk pk −1
pk −1
|uk (x)| (ξk
rk (s)|u0k (s)|pk ds (2.17)
(x) + ηk
(x)) ≤ (ξk (x)ηk (x))
a
for k = 1, 2, . . . , n and x ∈ [a, b]. It is easy to see that the functions ξkpk −1 (x) +
ηkpk −1 (x) take the minimum values at ck ∈ (a, b) such that ξk (ck ) = ηk (ck ) for
k = 1, 2, . . . , n. Thus, we obtain
Z b
pk −1
pk pk −1
pk −1
|uk (x)| (ξk
(ck ) + ηk
(ck )) ≤ (ξk (x)ηk (x))
rk (s)|u0k (s)|pk ds (2.18)
a
for k = 1, 2, . . . , n. Since ξk (ck ) + ηk (ck ) = ξk (x) + ηk (x), ∀x, ck ∈ (a, b), and
Z
1 b 1/(1−pk )
ξk (x) + ηk (x)
=
r
(s)ds,
ξk (ck ) =
2
2 a k
we have
|uk (x)|pk [22−pk (ξk (x) + ηk (x))pk −1 ]
= |uk (x)|pk [2ξkpk −1 (ck )]
Z b
pk −1
≤ (ξk (x)ηk (x))
rk (s)|u0k (s)|pk ds
(2.19)
a
and hence
|uk (x)|pk ≤ Ek (x)
Z
b
rk (s)|u0k (s)|pk ds
(2.20)
a
for k = 1, 2, . . . , n and x ∈ [a, b]. After that by using a technique similar to the one
in [9, Theorem 3.1], it can be showed that the equality case in (2.20) does not hold.
Thus, we have
|uk (x)|pk < Ak Ek (x), x ∈ (a, b),
(2.21)
Rb
where Ak = a rk (s)|u0k (s)|pk ds for k = 1, 2, . . . , n. The rest of the proof is the
same as in the proof of Theorem 2.1, and hence is omitted.
Remark 2.3. It is easy to see from the inequality (1.13) that if we take 1 < pk < 2
for k = 1, 2, . . . , n, then inequality (2.14) is better than (2.2) in the sense that (2.2)
follows from (2.14), but not conversely. Similarly, from the inequality (1.14), if
pk > 2 for k = 1, 2, . . . , n, then inequality (2.2) is better than (2.14) in the sense
that (2.14) follows from (2.2), but not conversely.
By using the inequality (1.15) in Theorem 2.1 or (1.17) in Theorem 2.2, we
obtain the following result.
EJDE-2013/67
LYAPUNOV-TYPE INEQUALITIES
7
Corollary 2.4. Assume that there exist nontrivial solutions (e1 , e2 , . . . , en ) of system (2.1). If fk ∈ C([a, b], R) for k = 1, 2, . . . , n and (u1 (x), u2 (x), . . . , un (x)) is a
nontrivial solution on [a, b] for system (1.1) with αki = αik for k, i = 1, 2, . . . , n,
then
Z b
n ek
Y
1<
Fk
fk+ (s)ds
.
(2.22)
k=1
a
Remark 2.5. Note that Theorem 2.1 or 2.2 yields a new Lyapunov-type inequality
which is not covered by Theorem 1.4 given by Yang et al [12]. It is easy to see
that Corollary 2.4 coincides with Theorem 1.4 under the condition αki = αik for
k, i = 1, 2, . . . , n.
Remark 2.6. Since |f (x)| ≥ f + (x), the functions fk+ (x) for in the above results
can also be replaced by |fk (x)| for k = 1, 2, . . . , n.
Now, we give an application of the obtained Lyapunov-type inequalities for the
eigenvalue problem
−(rk (x)φpk (u0k ))0 = λk h(x)φαkk (uk )
n
Y
|ui |αki
i=1, i6=k
(2.23)
uk (a) = uk (b) = 0,
where h(x) > 0. Thus, if there exist nontrivial solutions (e1 , e2 , . . . , en ) of linear
homogeneous system (2.1), then we have
n hZ b
n
n n−1
iek o− e1
Y e Y
Y
n
α /p
< λn
λkk
h(s)
Di ki i (s)ds
k=1
a
k=1
i=1
or
n n−1
Y
k=1
λekk
n hZ
Y
k=1
a
b
h(s)
n
Y
α
Ei ki
/pi
iek o− e1
n
(s)ds
< λn .
i=1
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Mustafa Fahri Aktaş
Gazi University, Faculty of Sciences, Department of Mathematics, 06500 Teknikokullar,
Ankara, Turkey
E-mail address: mfahri@gazi.edu.tr