AbstractWe state and prove a generalized Lyapunov-type inequality for one-dimensional Dirichlet quasilinear systems involving the (p1,p2,…,pn)-Laplacian. Our result generalize the Lyapunov-type inequality given in Napoli and Pinasco (2006) [12]

Tiryaki, Aydın

Çakmak, Devrim

Elsevier - Publisher Connector 

Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the (p1,p2,…,pn)-Laplacian 

J. Math. Anal. Appl. 369 (2010) 76–81Contents lists available at ScienceDirect
Journal of Mathematical Analysis and
Applications
www.elsevier.com/locate/jmaa
Lyapunov-type inequality for a class of Dirichlet quasilinear systems
involving the (p1, p2, . . . , pn)-Laplacian
Devrim Çakmak a,∗, Aydın Tiryaki b
a Gazi University, Faculty of Education, Department of Mathematics Education, 06500 Teknikokullar, Ankara, Turkey
b Izmir University, Faculty of Arts and Sciences, Department of Mathematics and Computer Sciences, 35350 Uckuyular, Izmir, Turkey
a r t i c l e i n f o a b s t r a c t
Article history:
Received 27 January 2010
Available online 24 February 2010
Submitted by J.S.W. Wong
Keywords:
Lyapunov-type inequality
Quasilinear system
(p1, p2, . . . , pn)-Laplacian
Lower bound
Half-linear differential equation
We state and prove a generalized Lyapunov-type inequality for one-dimensional Dirichlet
quasilinear systems involving the (p1, p2, . . . , pn)-Laplacian. Our result generalize the
Lyapunov-type inequality given in Napoli and Pinasco (2006) [12].
© 2010 Elsevier Inc. All rights reserved.
1. Introduction
The Lyapunov inequality and many of its generalizations have proven to be useful tools in oscillation theory, disconjugacy,
eigenvalue problems, and numerous other applications for the theories of differential and difference equations. For authors,
who contributed to the Lyapunov-type inequalities, we refer to Brown and Hinton [2], Cheng [3], Çakmak [4], Došlý and
R˘ehák [5], Hartman [7], Kwong [9], Lee et al. [10], Liapunov [11], Pachpatte [13–15], Panigrahi [16], Parhi and Panigrahi
[17,18], Pinasco [20], Yang and Lo [24], and the references quoted therein.
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. Recently, Guseinov and Kaymakçalan [6], and Tiryaki, Ünal and
Çakmak [21] have obtained the Lyapunov-type inequalities for ﬁrst order systems. The discrete and time scale analogues of
Lyapunov-type inequalities for certain type systems are also given by Ünal, Çakmak and Tiryaki [22], Jiang and Zhou [8], and
Ünal and Çakmak [23].
More recently, Napoli and Pinasco [12] have interested in the problem of ﬁnding the Lyapunov-type inequality for the
following quasilinear systems involving (p,q)-Laplacian operators
−(∣∣u′∣∣p−2u′)′ = f1(x)|u|α−2u|v|β
−(∣∣v ′∣∣q−2v ′)′ = f2(x)|u|α |v|β−2v
⎫⎬
⎭ (1)
where f1, f2 are real-valued continuous functions for all x ∈ R, the exponents satisfy 1 < p,q < ∞, and the positive
parameters α, β satisfy
α
p
+ β
q
= 1. (2)
Their results are as follows:
* Corresponding author.
E-mail addresses: dcakmak@gazi.edu.tr (D. Çakmak), aydin.tiryaki@izmir.edu.tr (A. Tiryaki).0022-247X/$ – see front matter © 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2010.02.043
D. Çakmak, A. Tiryaki / J. Math. Anal. Appl. 369 (2010) 76–81 77Theorem A. If system (1) with fi(x) > 0 for i = 1,2 has a real nontrivial solution (u(x), v(x)) such that u(a) = u(b) = 0 = v(a) =
v(b) where a,b ∈ R with a < b be consecutive zeros, and u and v are not identically zero on [a,b], then the inequality
2α+β  (b − a)α+β−1
( b∫
a
f1(x)dx
)α/p( b∫
a
f2(x)dx
)β/q
(3)
holds.
This result was used by proving the following theorem which improves the lower bounds on the eigenvalues of the
problem
−(∣∣u′∣∣p−2u′)′ = λαr(x)|u|α−2u|v|β
−(∣∣v ′∣∣q−2v ′)′ = μβr(x)|u|α |v|β−2v
⎫⎬
⎭ (4)
where r(x) be a positive function for all x ∈ R.
Theorem B. There exists a function h(λ) such that μ  h(λ) for every generalized eigenvalue (λ,μ) of problem (4), where h(λ) is
given by
h(λ) = 1
β
(
C
λα/p
∫ b
a r(x)dx
)q/β
, (5)
and the constant C is given by
C = 2
α+β
αα/p(b − a)α+β−1 . (6)
In this paper, by using a similar technique to that of Napoli and Pinasco [12], we state and prove a generalized Lyapunov-
type inequality for a Dirichlet problem associated to the following quasilinear systems involving (p1, p2, . . . , pn)-Laplacian
operators
−(∣∣u′1∣∣p1−2u′1)′ = f1(x)|u1|α1−2u1|u2|α2 . . . |un|αn
−(∣∣u′2∣∣p2−2u′2)′ = f2(x)|u1|α1 |u2|α2−2u2|u3|α3 . . . |un|αn
. . .
−(∣∣u′n∣∣pn−2u′n)′ = fn(x)|u1|α1 . . . |un−1|αn−1 |un|αn−2un
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
(7)
where n ∈ N, f i are real-valued continuous functions for all x ∈ R, the exponents satisfy 1 < pi < ∞ and the positive
parameters αi satisfy
n∑
i=1
αi
pi
= 1 (8)
for i = 1,2, . . . ,n. If n = 1, then system (7) reduces to the following equation(∣∣u′1∣∣p1−2u′1)′ + f1(x)|u1|p1−2u1 = 0 (9)
which is known as the half linear equation. Similarly, if n = 2, then system (7) reduces to the system
−(∣∣u′1∣∣p1−2u′1)′ = f1(x)|u1|α1−2u1|u2|α2
−(∣∣u′2∣∣p2−2u′2)′ = f2(x)|u1|α1 |u2|α2−2u2
⎫⎬
⎭ (10)
which is the same system as (1).
The aim of this paper is to extend and generalize Theorems A and B of Napoli and Pinasco [12] to the general case. Our
motivation comes from the recent paper of Afrouzi and Heidarkhani [1].
We derive a Lyapunov-type inequality for quasilinear system (7), where all components of the solution (u1(x),u2(x), . . . ,
un(x)) have consecutive zeros at the points a,b ∈ R with a < b in I = [t0,∞) ⊂ R. For the special cases of system (7), we
also derive some Lyapunov-type inequalities which relates not only points a and b in I at which all components of the
solution (u1(x),u2(x), . . . ,un(x)) have consecutive zeros but also any point in (a,b) where all components of the solution
(u1(x),u2(x), . . . ,un(x)) are maximized.
Since our attention is restricted to the Lyapunov-type inequality for the quasilinear system of differential equations, we
shall assume the existence of the nontrivial solution (u1(x),u2(x), . . . ,un(x)) of system (7).
78 D. Çakmak, A. Tiryaki / J. Math. Anal. Appl. 369 (2010) 76–812. Main results
The main result of this paper is the following theorem:
Theorem 1. If system (7) has a real nontrivial solution (u1(x),u2(x), . . . ,un(x)) such that ui(a) = 0 = ui(b) for i = 1,2, . . . ,n where
n ∈ N, a,b ∈ R with a < b are consecutive zeros and ui for i = 1,2, . . . ,n are not identically zero on [a,b], then the inequality
n∏
i=1
{
(ci − a)1−pi + (b − ci)1−pi
}αi/pi  n∏
i=1
( b∫
a
f +i (x)dx
)αi/pi
(11)
holds, where |ui(ci)| = maxa<x<b |ui(x)| and f +i (x) = max{0, f i(x)} for i = 1,2, . . . ,n.
Proof. It follows from ui(a) = 0 = ui(b) for i = 1,2, . . . ,n where n ∈ N, a,b ∈ R with a < b are consecutive zeros and ui
for i = 1,2, . . . ,n are not identically zero on [a,b], one can choose ci ∈ (a,b) such that |ui(ci)| = maxa<x<b |ui(x)| > 0 for
i = 1,2, . . . ,n. From Rolle’s theorem, clearly u′i(ci) = 0 for i = 1,2, . . . ,n. Therefore, for c1 ∈ (a,b) and u1(a) = 0, we have
∣∣u1(c1)∣∣=
∣∣∣∣∣
c1∫
a
u′1(x)dx
∣∣∣∣∣
c1∫
a
∣∣u′1(x)∣∣dx. (12)
By using Hölder inequality on the integral of the right-hand side of (12) with indices p1 and p′1, we obtain
∣∣u1(c1)∣∣
c1∫
a
∣∣u′1(x)∣∣dx (c1 − a) 1p′1
( c1∫
a
∣∣u′1(x)∣∣p1 dx
) 1
p1
(13)
where 1p1 + 1p′1 = 1. On the other hand, multiplying the ﬁrst equation of system (7) by u1 and integrating from a to c1 and
taking into account that u1(a) = 0 and u′1(c1) = 0, we get
c1∫
a
∣∣u′1(x)∣∣p1 dx =
c1∫
a
f1(x)
∣∣u1(x)∣∣α1 ∣∣u2(x)∣∣α2 . . . ∣∣un(x)∣∣αn dx. (14)
Therefore, by using (14) in (13), we have
∣∣u1(c1)∣∣ (c1 − a) 1p′1
( c1∫
a
f1(x)
∣∣u1(x)∣∣α1 ∣∣u2(x)∣∣α2 . . . ∣∣un(x)∣∣αn dx
) 1
p1
 (c1 − a)
1
p′1
( c1∫
a
f +1 (x)
∣∣u1(x)∣∣α1 ∣∣u2(x)∣∣α2 . . . ∣∣un(x)∣∣αn dx
) 1
p1
. (15)
If we take the p1-th power of both sides of inequality (15), and |ui(x)| is maximum at the point ci for i = 1,2, . . . ,n,
respectively, we obtain
1
∣∣u1(c1)∣∣α1−p1 ∣∣u2(c2)∣∣α2 . . . ∣∣un(cn)∣∣αn(c1 − a)p1−1
( c1∫
a
f +1 (x)dx
)
. (16)
Now, since u1(b) = 0, we get
∣∣u1(c1)∣∣= ∣∣−u1(c1)∣∣=
∣∣∣∣∣
b∫
c1
u′1(x)dx
∣∣∣∣∣
b∫
c1
∣∣u′1(x)∣∣dx (17)
and repeating the above procedure step by step, one can easily obtain
1
∣∣u1(c1)∣∣α1−p1 ∣∣u2(c2)∣∣α2 . . . ∣∣un(cn)∣∣αn(b − c1)p1−1
( b∫
c1
f +1 (x)dx
)
. (18)
D. Çakmak, A. Tiryaki / J. Math. Anal. Appl. 369 (2010) 76–81 79Thus, summing up inequalities (16) and (18), we have
b∫
a
f +1 (x)dx
∣∣u1(c1)∣∣p1−α1 ∣∣u2(c2)∣∣−α2 . . . ∣∣un(cn)∣∣−αn{(c1 − a)1−p1 + (b − c1)1−p1}. (19)
By using similar manner, we get the following inequalities
b∫
a
f +2 (x)dx
∣∣u1(c1)∣∣−α1 ∣∣u2(c2)∣∣p2−α2 ∣∣u3(c3)∣∣−α3 . . . ∣∣un(cn)∣∣−αn{(c2 − a)1−p2 + (b − c2)1−p2}, (20)
. . .
b∫
a
f +n (x)dx
∣∣u1(c1)∣∣−α1 . . . ∣∣un−1(cn−1)∣∣−αn−1 ∣∣un(cn)∣∣pn−αn{(cn − a)1−pn + (b − cn)1−pn}, (21)
respectively. Raising inequality (19) to a power e1, inequality (20) to a power e2, . . . , and inequality (21) to a power en , and
multiplying the resulting equations, we obtain
n∏
i=1
( b∫
a
f +i (x)dx
)ei

[
n∏
i=1
{
(ci − a)1−pi + (b − ci)1−pi
}ei]
× ∣∣u1(c1)∣∣(p1−α1)e1−α1e2−···−α1en
× ∣∣u2(c2)∣∣−α2e1+(p2−α2)e2−α2e3−···−α2en
× · · · × ∣∣un(cn)∣∣−αne1−···−αnen−1+(pn−αn)en
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(22)
for i = 1,2, . . . ,n. Now, we choose ei such that |ui(ci)| cancel out for i = 1,2, . . . ,n in inequality (22), i.e. solve the homo-
geneous linear system
(p1 − α1)e1 − α1e2 − · · · − α1en = 0
−α2e1 + (p2 − α2)e2 − α2e3 − · · · − α2en = 0
. . .
−αne1 − · · · − αnen−1 + (pn − αn)en = 0.
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
(23)
We observe that by hypothesis
∑n
i=1
αi
pi
= 1, this system admits a nontrivial solution, indeed all equations are equivalent to
ei
(
n∑
j=1
j =i
α j
p j
)
= αi
pi
(
n∑
j=1
j =i
e j
)
(24)
for i = 1,2, . . . ,n. Hence, we may take ei = αipi for i = 1,2, . . . ,n, and we get inequality (11) which completes the proof. 
Remark 2. The left-hand side of inequality (11) shows that ci for i = 1,2, . . . ,n cannot be too close to a or b, since the
exponents satisfy 1 < pi < ∞ for i = 1,2, . . . ,n. We have
∫ b
a f
+
i (x)dx < ∞ for i = 1,2, . . . ,n, but
lim
ci→a+
ci→b−
{
(ci − a)1−pi + (b − ci)1−pi
}= ∞ (25)
for i = 1,2, . . . ,n.
Since the function h(x) = x1−pi is convex for x > 0 and pi > 1 for i = 1,2, . . . ,n, Jensen’s inequality h( y+z2 ) 
1
2 [h(y) + h(z)] with y = ci − a and z = b − ci for i = 1,2, . . . ,n implies
(ci − a)1−pi + (b − ci)1−pi  2pi (b − a)1−pi (26)
for i = 1,2, . . . ,n. Thus, by using inequality (26), Theorem 1 reduces to the following result:
80 D. Çakmak, A. Tiryaki / J. Math. Anal. Appl. 369 (2010) 76–81Corollary 3. If system (7) has a real nontrivial solution (u1(x),u2(x), . . . ,un(x)) such that ui(a) = 0 = ui(b) for i = 1,2, . . . ,n where
n ∈ N, a,b ∈ R with a < b are consecutive zeros and ui for i = 1,2, . . . ,n are not identically zero on [a,b], then the inequality
2(
∑n
i=1 αi)(b − a)1−(
∑n
i=1 αi) 
n∏
i=1
( b∫
a
f +i (x)dx
)αi/pi
(27)
holds, where |ui(ci)| = maxa<x<b |ui(x)| and f +i (x) = max{0, f i(x)} for i = 1,2, . . . ,n.
Remark 4. Let n = 1. If we compare inequalities (16) and (18) with the inequalities of (2.3) in Lemma 2.1 of Pinasco [19]
respectively, it is easy to see that the restricted condition, i.e. a bounded positive function, on the function r in Lemma 2.1
can be dropped. Thus, Theorem 1 (or Corollary 3) generalizes and extends Theorem 2.3 of Pinasco [19].
Remark 5. Let n = 2. If we compare Theorem 1 with Theorem A of Napoli and Pinasco [12], since (26) holds, we conclude
that Theorem 1 is more general than Theorem A.
Remark 6. Corollary 3 with n = 2 and f i(x) > 0 for i = 1,2 reduces to Theorem A.
Remark 7. When αi = pi for i = 1,2, . . . ,n, and for j = i, α j = 0 for j = 1,2, . . . ,n, we obtain the result for the case of
a single equation from Theorem 1 or Corollary 3.
Remark 8. Since
f +(x)
∣∣ f (x)∣∣, (28)
the integrals of
∫ b
a f
+
i (x)dx for i = 1,2, . . . ,n in the above results can also be replaced by
∫ b
a | f i(x)|dx for i = 1,2, . . . ,n,
respectively.
Now, we present an application of the obtained Lyapunov-type inequality for system (7).
Let λi for i = 1,2, . . . ,n be generalized eigenvalues of system (7), and r(x) be a positive function for all x ∈ R. Therefore,
system (7) with f i(x) = λiαir(x) > 0 for i = 1,2, . . . ,n and all x ∈ R reduces to the following system:
−(∣∣u′1∣∣p1−2u′1)′ = λ1α1r(x)|u1|α1−2u1|u2|α2 . . . |un|αn
−(∣∣u′2∣∣p2−2u′2)′ = λ2α2r(x)|u1|α1 |u2|α2−2u2|u3|α3 . . . |un|αn
. . .
−(∣∣u′n∣∣pn−2u′n)′ = λnαnr(x)|u1|α1 . . . |un−1|αn−1 |un|αn−2un.
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
(29)
By using similar techniques to the technique in Napoli and Pinasco [12], we obtain the following result which gives lower
bounds for the n-th eigenvalue of λn . The proof of the following theorem is based on above generalization of the Lyapunov-
type inequality, as in that of Theorem 1.4 of Napoli and Pinasco [12].
Theorem 9. There exists a function h1(λ1, λ2, . . . , λn−1) such that λn  h1(λ1, λ2, . . . , λn−1) for every generalized eigenvalue
(λ1, λ2, . . . , λn) of the quasilinear system (29), where |ui(ci)| = maxa<x<b |ui(x)| for i = 1,2, . . . ,n and
h1(λ1, λ2, . . . , λn−1) = 1
αn
(∏n
i=1{(ci − a)1−pi + (b − ci)1−pi }αi/pi
(
∏n−1
i=1 (λiαi)αi/pi )(
∫ b
a r(x)dx)
)pn/αn
. (30)
Proof. Let (λ1, λ2, . . . , λn) be a generalized eigenpair, and u1,u2, . . . ,un be the corresponding nontrivial solution of sys-
tem (29). For i = 1,2, . . . ,n and all x ∈ R, by substituting λiαir(x) > 0 for f i(x) in the Lyapunov inequality (11), we obtain
n∏
i=1
{
(ci − a)1−pi + (b − ci)1−pi
}αi/pi  n∏
i=1
( b∫
a
λiαir(x)dx
)αi/pi
. (31)
Rearranging the terms, and by using condition
∑n
i=1
αi
pi
= 1, we obtain
n∏
i=1
{
(ci − a)1−pi + (b − ci)1−pi
}αi/pi 
(
n∏
i=1
(λiαi)
αi/pi
)( b∫
r(x)dx
)
, (32)a
D. Çakmak, A. Tiryaki / J. Math. Anal. Appl. 369 (2010) 76–81 81which gives(∏n
i=1{(ci − a)1−pi + (b − ci)1−pi }αi/pi
(
∏n−1
i=1 (λiαi)αi/pi )(
∫ b
a r(x)dx)
)pn/αn
 λnαn, (33)
and the proof is completed. 
Remark 10. Since h1 is a continuous function, then h1(λ1, λ2, . . . , λn−1) → +∞ as any eigenvalue of λi → 0+ for i =
1,2, . . . ,n − 1. Therefore, there exists a ball centered in the origin such that the generalized spectrum is contained in its
exterior. Also, by rearranging terms in (33) we obtain
n∏
i=1
λ
αi/pi
i 
(∏n
i=1{(ci − a)1−pi + (b − ci)1−pi }αi/pi
(
∏n
i=1 α
αi/pi
i )(
∫ b
a r(x)dx)
)
. (34)
It is clear that when the interval collapses, right-hand side of (34) goes to inﬁnity. Hence, we obtain the desired generaliza-
tions of Napoli and Pinasco [12]’s result for one-dimensional nonlinear systems.
Remark 11. Let n = 1. If we compare inequality (34) with inequality (1.2) with n = 1 in Theorem 1.1 of Pinasco [19], it is
easy to see that inequality (34) gives better lower bound than inequality (1.2) with n = 1. Thus, Theorem 9 generalizes and
extends Theorem 1.1 with n = 1 of Pinasco [19].
Remark 12. Let n = 2. If we compare Theorem 9 with Theorem B of Napoli and Pinasco [12], we obtain h1(λ1) h(λ) since
(26) holds. Therefore, Theorem 9 gives better lower bound than Theorem B.
Remark 13. Let n = 2. Theorem 9 with c1 = a+b2 = c2 or with inequality (26) reduces to Theorem B.
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Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the (p1,p2,…,pn)-Laplacian

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