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Lyapunov-type inequalities for 2M th order equations under clamped-free boundary conditions

  • 1Email author,
  • 2,
  • 3,
  • 2 and
  • 4
Journal of Inequalities and Applications20122012:242

https://doi.org/10.1186/1029-242X-2012-242

  • Received: 29 May 2012
  • Accepted: 9 October 2012
  • Published:

Abstract

This paper generalizes the well-known Lyapunov-type inequalities for second-order linear differential equations to certain 2M th order linear differential equations

( 1 ) M u ( 2 M ) ( x ) r ( x ) u ( x ) = 0 ( s x s )

under clamped-free boundary conditions. The usage of the best constant of some kind of a Sobolev inequality helps clarify the process for obtaining the result.

Keywords

  • Green Function
  • Classical Solution
  • Linear Differential Equation
  • Sobolev Inequality
  • Distributional Sense

1 Introduction

Let us consider the second-order linear differential equation
{ u ( x ) + r ( x ) u ( x ) = 0 ( s x s ) , u ( ± s ) = 0 ,
(1)
where . It is well known that the Lyapunov inequality
s s r + ( x ) d x > 2 s
(2)
gives a necessary condition for the existence of non-trivial classical solutions of (1), where r + ( x ) = ( r ( x ) + | r ( x ) | ) / 2 . There are various extensions and applications for the above result; see, for example, surveys of Brown and Hinton [1] for relations to other fields and Tiryaki [2] for recent developments. Extensions to higher-order equations
{ u ( n ) ( x ) + r ( x ) u ( x ) = 0 ( s x s ) , Boundary Conditions ,
(3)

will be one important aspect. The first result for the high-order equation (3) is due to Levin [3], which states without proof:

Theorem A Let n = 2 M , and a non-trivial solution of (3) satisfies the clamped boundary condition, u ( i ) ( ± s ) = 0 ( i = 0 , 1 , , M 1 ). Then it holds that
s s r + ( x ) d x > 2 2 M 1 ( 2 M 1 ) { ( M 1 ) ! } 2 s 2 M 1 .
Later, Das and Vatsala [4] gave the proof and extended the result by constructing the Green function. Other interesting developments for higher-order equations are seen in [59]. For example, as shown in Yang [8], Lyapunov-type inequalities can be obtained under the following conditions:
{ ( a ) n = 2 M + 1 , u ( i ) ( ± s ) = 0 ( i = 0 , , M 1 ) , u ( 2 M ) ( d ) = 0 ( s < d < s )  (see also [10]) ( b ) n 2 , u ( s ) = u ( t 2 ) = = u ( t n 1 ) = u ( s ) = 0 , where,  s = t 1 < t 2 < < t n 1 < t n = s  (see also [10]) ( c ) n = 2 M , u ( 2 i ) ( ± s ) = 0 ( i = 0 , , M 1 )  (see also [5]) ( d ) n 2 , u ( i ) ( s ) = 0 ( i = 0 , , k 1 ) , u ( j ) ( s ) = 0 ( j = 0 , , n k 1 ) where  k  runs over the range  ( 1 k n ) .
Here we note for the condition (c), very recently Çakmak [11], He and Tang [12], He and Zang [13] and [14] improved and extended the results of [5] and [8]. This paper considers the necessary condition for the existence of a non-trivial solution of the 2M th order linear differential equation
( 1 ) M u ( 2 M ) ( x ) r ( x ) u ( x ) = 0 ( s x s )
(4)

under yet another boundary condition:

Clamped-free boundary condition
u ( i ) ( s ) = 0 , u ( M + i ) ( s ) = 0 ( i = 0 , , M 1 ) .

The main result is as follows.

Theorem 1 Suppose a non-trivial solution u of (4) exists under the clamped-free boundary condition, then it holds
s s r + ( x ) d x > { ( M 1 ) ! } 2 ( 2 M 1 ) ( 2 s ) 2 M 1 .
(5)

Moreover, the estimate is sharp in the sense that there exists a function r ( x ) , and for this r ( x ) , the solution u of (4) exits such that the right-hand side is arbitrarily close to the left-hand side.

The result is obtained using Takemura [[15], Theorem 1], which computes the best constant of some kind of a Sobolev inequality. In Section 4, we give a concise proof for an L p extension of Theorem 1 of [15].

2 Proof of Theorem 1

Now, let us introduce the following L p -type Sobolev inequality:
( sup s x s | u ( m ) ( x ) | ) p C s s | u ( M ) ( x ) | p d x ,
(6)
where u belongs to
W ( M , p ) : = { u | u ( M ) L p ( s , s ) , u ( i ) ( s ) = 0 ( i = 0 , , M 1 ) } ,

1 < p , m runs over the range 0 m M 1 , and u ( i ) is the i th derivative of u in a distributional sense. We denote by C C F ( M , m , p ) the best constant of the above Sobolev inequality (6). Here, we note that in [15], Takemura obtained the best constant for p = 2 , m = 0 by constructing the Green function of the clamped-free boundary value problem. Although, for the proof of Theorem 1, we simply need the value C C F ( M , 0 , 2 ) , we would like to compute C C F ( M , m , p ) for general p and m since the proof presented in Section 4 does not depend on special values of p and m and quite simplifies the proof of Theorem 1 of [15]. Now, we have the following propositions.

Proposition 1 The best constant of (6) is
C C F ( M , m , p ) = 1 { ( M m 1 ) ! } p ( ( p 1 ) ( 2 s ) p ( M m ) 1 p 1 p ( M m ) 1 ) p 1 ,
(7)
and it is attained by
u ( x ) = s x ( x t ) M 1 ( M 1 ) ! { ( s t ) M 1 m ( M 1 m ) ! } p 1 d t .
(8)
Proposition 2 Suppose a C 2 M [ s , s ] solution of (4) with the clamped-free boundary condition exists, then it holds that
s s r + ( x ) d x > 1 C C F ( M , 0 , 2 ) .
(9)

Moreover, the estimate is sharp.

Proof of Theorem 1 Clearly, Theorem 1 is obtained from Propositions 1 and 2. □

Thus, all we have to do is to show Propositions 1 and 2. Before proceeding with the proof of these propositions, we would like to show a corollary obtained from Proposition 1.

Corollary 1 Suppose a non-trivial solution u of the non-linear equation
( 1 ) M u ( x ) u ( 2 M ) ( x ) r ( x ) ( u ( m ) ( x ) ) 2 = 0
(10)
exists under the clamped-free boundary condition, where m satisfies ( 1 m M 1 ), then it holds
s s r + ( x ) d x > { ( M 1 m ) ! } 2 ( 2 ( M m ) 1 ) ( 2 s ) 2 ( M m ) 1 .
(11)

The following are the examples of Theorem 1 and Corollary 1.

Example 1 The following example corresponds to the case M = 1 and r ( x ) = 6 / ( 11 s 2 + 2 s x + x 2 ) of (4) with the clamped-free boundary condition
{ u ( x ) + 6 11 s 2 + 2 s x + x 2 u ( x ) = 0 , u ( s ) = u ( s ) = 0 .
It is easy to see that u ( x ) = ( s + x ) ( 11 s 2 2 s x x 2 ) is the solution of the above equation. Moreover, it holds that
s s r + ( x ) d x = s s 6 11 s 2 + 2 s x + x 2 d x = 3 log ( 2 + 3 ) 2 s > 1 C C F ( 1 , 0 , 2 ) = 1 2 s .
Example 2 The following example corresponds to the case M = 2 , m = 1 and r ( x ) = ( 3 ( 17 s 2 6 s x + x 2 ) ) / ( 2 ( 7 s 2 4 s x + x 2 ) 2 ) of (10) with the clamped-free boundary condition
{ u ( x ) u ( 4 ) ( x ) 3 ( 17 s 2 6 s x + x 2 ) 2 ( 7 s 2 4 s x + x 2 ) 2 ( u ( x ) ) 2 = 0 , u ( s ) = u ( s ) = u ( s ) = u ( s ) = 0 .
It is easy to see that u ( x ) = ( s + x ) 2 ( 17 s 2 6 s x + x 2 ) is the solution of the above equation. Moreover, it holds that
s s r + ( x ) d x = s s 3 ( 17 s 2 6 s x + x 2 ) 2 ( 7 s 2 4 s x + x 2 ) 2 d x = 3 + 2 3 π 12 s > 1 C C F ( 2 , 1 , 2 ) = 1 2 s .

3 Proof of Proposition 2

Assuming Proposition 1, we first prove Proposition 2.

Proof of Proposition 2 Let u be a solution of equation (4). Since u satisfies the clamped-free boundary condition, multiplying (4) by u and integrating it over [ s , s ] , we have
s s ( u ( M ) ( x ) ) 2 d x = s s ( 1 ) M u ( x ) u ( 2 M ) ( x ) d x = s s r ( x ) ( u ( x ) ) 2 d x ( sup s x s | u ( x ) | ) 2 s s r + ( x ) d x C ( M , 0 , 2 ) s s ( u ( M ) ( x ) ) 2 d x s s r + ( x ) d x .
(12)
Here, if u ( M ) 0 , then there exists ( i = 0 , , M 1 ) such that u ( x ) = i = 0 M 1 a i x i . Since u satisfies the clamped boundary condition at x = s , we have u 0 . This contradicts the assumption that u is a non-trivial solution of (4). So, canceling u ( M ) 2 2 , we obtain
s s r + ( x ) d x 1 C C F ( M , 0 , 2 ) .
(13)
Next, we show that the inequality (13) is strict. To see this, we note that in (12), if the equality holds for the first inequality, then u is a constant. But, again from the clamped boundary condition at x = s , we have u 0 . Thus, the inequality is strict. Finally, we see (5) is sharp. For this purpose, let us define the functional
J ( ϕ ) : = s s | ϕ ( M ) | 2 d x s s r ˜ | ϕ | 2 d x ( ϕ W ( M , 2 ) , ϕ 0 ) ,
where is defined later. By the standard argument of the variational method, J has the minimizer u W ( M , 2 ) (see, for example, [[16], Lemma 3]), i.e.,
λ 1 : = min ϕ W ( M , 2 ) , ϕ 0 J ( ϕ ) = J ( u ) .
Hence, it satisfies the Euler-Lagrange equation (as a classical solution by the regularity argument)
( 1 ) M u ( 2 M ) ( x ) = λ 1 r ˜ ( x ) u ( x ) ( s x s ) .
(14)
Further, it holds that
λ 1 = min ϕ W ( M , 2 ) , ϕ 0 s s | ϕ ( M ) | 2 d x s s r ˜ | ϕ | 2 d x > s s | ϕ ( M ) | 2 d x ( sup s x s | ϕ | ) 2 s s r ˜ d x 1 C C F ( M , 0 , 2 ) s s r ˜ d x .
(15)
Here, let us fix r ˜ as
r ˜ ( x ) : = { x s δ + 1 ( s δ < x s ) , 0 ( s x s δ ) .
For such r ˜ , let us substitute ϕ = u (of Proposition 1) into (15). It is easy to see that u takes its maximum at x = s , hence by taking δ sufficiently small, we see that the right-hand side of (15) can be arbitrarily close to the left-hand side, i.e., for a small positive ϵ 1 , λ 1 can be written as
λ 1 = 1 C C F ( M , 0 , 2 ) s s r ˜ d x + ϵ 1 .
(16)
Putting r = λ 1 r ˜ , we see from (14) a solution u of
( 1 ) M u ( 2 M ) ( x ) = r ( x ) u ( x ) ( s x s )
(17)
exists, and from (16) r satisfies
s s r ( x ) d x = 1 C ( M , 0 , 2 ) + ϵ 1 s s r ˜ d x = 1 C ( M , 0 , 2 ) + ϵ 2 .
(18)

Hence, (5) is sharp. □

Proof of Corollary 1 Integrating equation (10), we have
s s ( u ( M ) ( x ) ) 2 d x = s s r ( x ) ( u ( m ) ( x ) ) 2 d x ( sup s x s | u ( m ) ( x ) | ) 2 s s r + ( x ) d x C ( M , m , 2 ) s s ( u ( M ) ( x ) ) 2 d x s s r + ( x ) d x .
(19)
As in the proof of Proposition 2, by canceling u ( M ) 2 2 , we have
s s r + ( x ) d x 1 C C F ( M , m , 2 ) .

Next, we show that the inequality (11) is strict. To see this, we note that in (19), the equality holds for the first inequality if and only if u ( m ) is a constant. Hence, from the clamped boundary condition at x = s , we have u ( m ) 0 . So, there exists ( i = 0 , , m 1 ) such that u ( x ) = i = 0 m 1 a i x i . But, again from the clamped boundary condition at x = s , we have u 0 . Thus, inequality (11) is strict. □

4 Proof of Proposition 1

We prepare the following lemmas for the proof of Proposition 1.

Lemma 1 Suppose there exists a function u W ( M , p ) which attains the best constant C ( M , n , p ) of (6), then it holds that
max s x s | u ( m ) ( x ) | = | u ( m ) ( s ) | .
Proof Suppose it holds that
max s x s | u ( m ) ( x ) | = | u ( m ) ( a ) | ,
where a s . Further, let us define
u ˜ ( x ) : = { 0 ( s x a ) , u ( x + a s ) ( a x s ) .
Then it holds u ˜ W ( M , p ) and
max s x s | u ˜ ( m ) ( x ) | = max s x s | u ( m ) ( x ) | = | u ( m ) ( a ) |
and u ˜ ( M ) L p ( s , s ) < u ( M ) L p ( s , s ) . Hence,
C ( M , m , p ) = ( max s x s | u ( m ) ( x ) | ) p u ( M ) L p ( s , s ) < ( max s x s | u ˜ ( m ) ( x ) | ) p u ˜ ( M ) L p ( s , s ) .

This contradicts the assumption that C ( M , m , p ) is the best constant of (6). □

Lemma 2 Let
H m ( x ) : = ( 1 ) M 1 m ( x s ) M 1 m ( M 1 m ) ! ,
then for u W ( M , p ) it holds that
u ( m ) ( s ) = s s u ( M ) ( x ) H m ( x ) d x .
(20)

Proof Integrating by parts, we obtain the result. □

Proof of Proposition 1 From Lemma 2, we see that if the function attains the best constant C ( M , m , p ) , it belongs to W ( M , m , p ) W ( M , p ) :
W ( M , m , p ) = { u W ( M , p ) | max s x s | u ( m ) ( x ) | = | u ( m ) ( s ) | } .
Let u W ( M , m , p ) . Then applying Hölder’s inequality to (20), we have
max s x s | u ( m ) ( x ) | = | u ( m ) ( s ) | H m L q ( s , s ) u ( M ) L p ( s , s ) ,
(21)
where q satisfies 1 / p + 1 / q = 1 . Hence, if there exists the function u W ( M , m , p ) which attains the equality of (21), it holds that C ( M , m , p ) = H m L q ( s , s ) p . On the contrary, we see that the equality holds for (21) if and only if u satisfies
u ( M ) ( x ) = ( sgn H m ( x ) ) | H m ( x ) | q 1 .
(22)
It is easy to see that
u ( x ) = s x ( x t ) m 1 ( M 1 ) ! ( sgn H m ( t ) ) | H m ( t ) | q 1 d t = s x ( x t ) m 1 ( M 1 ) ! { ( s t ) M 1 m ( M 1 m ) ! } 1 p 1 d t
satisfies (22) and belongs to W ( M , m , p ) . Thus, we have shown C ( M , m , p ) = H m L q ( s , s ) p . Now, we compute H m L q ( s , s ) p . It is
H m L q ( s , s ) p = 1 { ( M 1 m ) ! } p { s s ( s x ) q ( M 1 m ) d x } p q = 1 { ( M 1 m ) ! } p { ( p 1 ) ( 2 s ) p ( M m ) 1 p 1 p ( M m ) 1 } p 1 .

This completes the proof. □

Declarations

Authors’ Affiliations

(1)
Department of Computer Science, National Defense Academy, 1-10-20 Hashirimizu, Yokosuka 239-8686, Japan
(2)
Liberal Arts and Basic Sciences, College of Industrial Technology, Nihon University, 2-11-1 Shinei, Narashino 275-8576, Japan
(3)
Graduate School of Mathematical Sciences, Faculty of Engineering Science, Osaka University, 1-3 Matikaneyamacho, Toyonaka 560-8531, Japan
(4)
Tokyo Metropolitan College of Industrial Technology, 1-10-40 Higashi-ooi, Shinagawa, Tokyo 140-0011, Japan

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© Watanabe et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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