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On Liapunov-type inequality for certain higher-order differential equations
Applied Mathematics and Computation 134 (2003) 307–317 www.elsevier.com/locate/amc
On Liapunov-type inequality for certain higher-order differential equations Xiaojing Yang Department of Mathematics, Tsinghua University, Beijing 100084, PR China
Abstract In this paper, we generalize the well-known Liapunov-type inequality for a class of third-order differential equations without damping force to 2n þ 1 order differential equations. Ó 2002 Published by Elsevier Science Inc. Keywords: Liapunov’s inequalities; Green’s functions; Higher-order equations
1. Introduction It is well known that if xðtÞ is a solution of x00 þ qðtÞx ¼ 0 satisfying xðaÞ ¼ xðbÞ ¼ 0 ða < bÞ and xðtÞ 6¼ 0 for t 2 ða; bÞ, then Z b jqðtÞj dt > 4=ðb aÞ:
ð1Þ
ð2Þ
a
Hartman [1] generalized (2) as follows: Z b ðt aÞðb tÞqþ ðtÞ dt > b a; a
where qþ ðtÞ ¼ maxfqðtÞ; 0g.
E-mail address: [email protected] (X. Yang). 0096-3003/02/$ - see front matter Ó 2002 Published by Elsevier Science Inc. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 2 8 5 - 5
ð3Þ
308
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
Generalizations or improvements of (2) have been obtained by many authors, see e.g. [2–4]. Recently Parhi and Panigrahi [5] established an inequality similar to (2) for third-order differential equation of the form x000 þ qðtÞx ¼ 0;
ð4Þ
where q 2 Cð½0; 1Þ; RÞ. Their results are: Theorem A. If there exists a d 2 ða; bÞ such that x00 ðdÞ ¼ 0, then Z b 2 jqðtÞj dt > 4=ðb aÞ :
ð5Þ
a
Theorem B. If x00 ðtÞ 6¼ 0, t 2 ða; bÞ, xðtÞ has three consecutive zeros a < b < a0 , then Z a0 2 jqðtÞj dt > 4=ða0 aÞ : ð6Þ a
In this paper, we generalize the inequalities of (2), (3), (5) and (6) to certain higher-order differential equations.
2. Main results Theorem 1. Let n 2 N , qðtÞ 2 Cð½a; bÞ. If there exists a d 2 ða; bÞ such that xð2nÞ ðdÞ ¼ 0; where xðtÞ is a solution of the following differential equation xð2nþ1Þ ðtÞ þ gðtÞxðtÞ ¼ 0
ð7Þ
satisfying xðiÞ ðaÞ ¼ xðiÞ ðbÞ ¼ 0; i ¼ 0; 1; . . . ; n 1;
xðtÞ 6¼ 0; t 2 ða; bÞ;
ð8Þ
then Z
b 2n
jqðtÞj dt > n!2ðnþ1Þ =ðb aÞ :
ð9Þ
a
Proof. Since if xðtÞ is a solution of (7), then xðtÞ is also a solution of (7), we can assume xðtÞ > 0, t 2 ða; bÞ. Since xðaÞ ¼ xðbÞ ¼ 0, there exists a c 2 ða; bÞ such that x0 ðcÞ ¼ 0 and xðcÞ ¼ maxa 6 t 6 b xðtÞ. Thus Z c Z c 0 < xðcÞ ¼ x0 ðtÞ dt 6 jx0 ðtÞj dt: ð10Þ a
a
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
309
Since x0 ðaÞ ¼ x0 ðcÞ ¼ 0; there is a t21 2 ða; cÞ such that x00 ðt21 Þ ¼ 0 and from x0 ðbÞ ¼ x0 ðcÞ ¼ 0; there is a t22 2 ðc; bÞ such that x00 ðt22 Þ ¼ 0. In this way, we see there exist ftjk gjk¼1 ða; bÞ such that xðjÞ ðtjk Þ ¼ 0; k ¼ 1; . . . ; j, j ¼ 1; 2; . . . ; n, n j jþ1 1 k and 0 < tjþ1 < tj1 < c < tjj < tjþ1 < b, j ¼ 1; . . . ; n. Hence there exist ftjþn gk¼1 k ða; bÞ such that xðnþjÞ ðtnþj Þ ¼ 0, k ¼ 1; . . . ; n j, j ¼ 1; . . . ; n 1. 0 From x ðaÞ ¼ 0, we have Z t x0 ðtÞ ¼ x00 ðsÞ ds: a
Hence jx0 ðtÞj 6
Z
t
jx00 ðsÞj ds;
t 2 ½a; c:
a
By (10), Z
c
jx0 ðtÞj dt 6
xðcÞ 6 a
Z cZ a
jx00 ðsÞj ds dt:
t
By using the equality Z cZ t Z cZ c f ðt; sÞ ds dt ¼ f ðt; sÞ dt ds; a
a
ð11Þ
a
a
ð12Þ
s
we obtain Z cZ
c 00
jx ðsÞj dt ds ¼
xðcÞ 6 a
s
Z
c
ðc sÞjx00 ðsÞj ds:
ð13Þ
a
Since for 1 6 k 6 n 1, Z t Z t ðkÞ ðkþ1Þ jx ðtÞj ¼ x ðsÞ ds 6 jxðkþ1Þ ðsÞj ds; a
a
it is easy to obtain Z
c
xðcÞ 6 a
ðc tÞn 1 ðnÞ jx ðtÞj dt: ðn 1Þ!
ð14Þ
k 1 From above analysis, there exist ftnþj gn j k¼1 , j ¼ 1; . . . ; n 1; and tn 2 ða; cÞ, n k tn 2 ðc; bÞ, such that xðtnþj Þ ¼ 0, k ¼ 1; . . . ; n j. Therefore Z t Z c jxðnÞ ðtÞj 6 jxðnþ1Þ ðsÞj ds 6 jxðnþ1Þ ðsÞj ds; t 2 ½a; c; tn1
a
310
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
and jxðnþ1Þ ðtÞj 6
Z
t
jxðnþ2Þ ðsÞj ds 6 1 tnþ1
Z
b
jxðnþ2Þ ðsÞj ds 6
a
6 6
Z
b
Z
Z
b
Z
b
Z
a
b
jxðnþ3Þ ðsÞj ds
a
b
jxð2nÞ ðsÞj ds dt1 dtm : a a |fflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflaffl}
ð15Þ
n 1
By assumption, there exists a d 2 ða; bÞ such that xð2nÞ ðdÞ ¼ 0. This gives Z t Z b jxð2nþ1Þ ðsÞj ds 6 jxð2nþ1Þ ðsÞj ds jxð2nÞ ðtÞj 6 d a Z b Z b jqðsÞxðsÞj ds 6 jqðsÞj ds xðcÞ: ð16Þ ¼ a
a
Combining (14)–(16), we obtain " #Z Z b ðb aÞn 1 ðc aÞ c n 1 ðc tÞ dt jqðtÞj dt xðcÞ 0 < xðcÞ 6 ðn 1Þ! a a Z b xðcÞ ðnþ1Þ ðn 1Þ ðc aÞ 6 ðb aÞ jqðtÞj dt; n! a which means Z b jqðtÞj dt P a
n! ðb aÞ
n 1
ðc aÞ
nþ1
ð17Þ
:
Similarly, we can prove that Z b n! jqðtÞj dt P : n 1 nþ1 ðb aÞ ðb cÞ a
ð18Þ
Since the function f ðuÞ ¼ u ðnþ1Þ is convex (f 00 ðuÞ > 0 for u > 0), we have 1 ðb cÞ
þ nþ1
1 ðc aÞ
nþ1
>
2
¼ b a nþ1 2
2nþ2 ðb aÞ
nþ1
this, together with (17) and (18), yields Z b n!2nþ1 jqðtÞj dt > : 2n ðb aÞ a Remark 1. Let n ¼ 1. We get the result of Theorem A [5], moreover, from (17) and (18), we see c cannot be very close to a as well as to b, which is also a generalization of Theorem 3 of [5].
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
311 N
Corollary 1. If xðtÞ is a solution of (7), which has N zeros ftk gk¼1 in the interval ½a; b, a 6 t1 < t2 < < tN 6 b, and for each k ¼ 1; . . . ; N 1, there exists a dk 2 ðtk ; tkþ1 Þ such that xð2nÞ ðdk Þ ¼ 0, then 2n Z b ðb aÞ 2nþ1 < jqðtÞj dt: ð19Þ ðN 1Þ n!2nþ1 a Proof. From Theorem 1, for k ¼ 1; . . . ; N 1, Z tkþ1 n!2nþ1 jqðtÞj dt > : 2n ðtkþ1 tk Þ tk Hence Z
b
jqðtÞj P
a
N 1 Z X k¼1
tkþ1
jqðtÞj dt > n!2nþ1
tk
N 1 X k¼1
1 ðtkþ1 tk Þ
2n
:
ð20Þ
Since f ðuÞ ¼ u 2n is convex for u > 0, we have for xk ¼ tkþ1 tk > 0, k ¼ 1; . . . ; N 1, ! N 1 N 1 X X f ðxk Þ > ðN 1Þf xk =ðN 1Þ ; k¼1
k¼1
that is N 1 X
1
k¼1
ðtkþ1 tk Þ2n
>
ðN 1Þ
2nþ1
ðtN t1 Þ2n
Hence Eqs. (20) and (21) give (19).
2nþ1
P
ðN 1Þ
ðb aÞ2n
:
ð21Þ
Remark 2. Let n ¼ 1. Then the result of (19) is sharper than the result of Theorem 6 of [5] for N P 3. Theorem 2. Let n 2 N , n P 2, qðtÞ 2 Cð½a; bÞ. If the differential equation xðnÞ þ qðtÞx ¼ 0
ð22Þ
has a solution xðtÞ satisfying the boundary value problem xðaÞ ¼ xðt2 Þ ¼ ¼ xðtn 1 Þ ¼ xðbÞ ¼ 0;
ð23Þ
where a ¼ t1 < t2 < < tn 1 < tn ¼ b and xðtÞ 6¼ 0, t 2 ðtk ; tkþ1 Þ, k ¼ 1; . . . ; n 1, then Z b ðn 2Þ!nn 1 jqðtÞj dt > : ð24Þ ðn 1Þn 2 ðb aÞn 1 a
312
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
Proof. Let Gðt; sÞ be Green’s function for the following boundary value problem xðnÞ ðtÞ ¼ 0, xðt1 Þ ¼ xðt2 Þ ¼ ¼ xðtn Þ ¼ 0. Then Z b xðtÞ ¼ Gðt; sÞqðsÞxðsÞ ds: ð25Þ a
Beesack [6] proved that jGðt; sÞj 6
n Y ðt tk Þ=½ðn 1Þ!ðb aÞ
ð26Þ
k¼1
and for t 2 ½a; b, jGðt; sÞj 6
n Y
n 1
jt tk j=½ðn 1Þ!ðb aÞ 6
k¼1 n 2
¼
ðn 1Þ ðb aÞ ðn 2Þ!nn 1
ðn 1Þ ðb aÞ ðn 1Þ!nn 1
n 1
n 1
:
Let jxðcÞj ¼ maxa 6 t 6 b jxðtÞj. Then from (25) and (26), we obtain Z b 0 < jxðcÞj 6 jGðt; sÞj jqðsÞj jxðsÞj ds a
Z < a
b
ðn 1Þn 2 ðb aÞn 1 jqðsÞj ds jxðcÞj ; ðn 2Þ!nn 1
which yields (24). Remark 3. Let n ¼ 3. We have Z b 9 jgðtÞj dt > ; 2 2ðb aÞ a which is sharper than the result of [5]. Lemma 1. Let us consider the differential equation (22). If xðtÞ is a nontrivial solution of (22) satisfying some two-point boundary value problem li ðxðaÞÞ ¼ 0;
l~i ðxðbÞÞ ¼ 0;
i ¼ 1; . . . ; m;
ð27Þ
and Green’s function Gðt; sÞ for the boundary value problem xðnÞ ¼ 0, li ðaÞ ¼ l~i ðbÞ ¼ 0, i ¼ 1; . . . ; m, satisfies jGðt; sÞj 6 jG1 ðt; tÞG1 ðs; sÞj;
t; s 2 ½a; b:
ð28Þ
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
313
Moreover, qðtÞ 6 0; xðtÞ 6¼ 0, t 2 ða; bÞ, then Z b jG21 ðt; tÞqðtÞj dt P 1:
ð29Þ
a
Proof. Multiplying both sides of (25) by jqðtÞjxðtÞ, then integrating over ½a; b we have Z b Z bZ b 2 jqðtÞjx ðtÞ dt 6 jGðt; sÞj jqðsÞj jxðsÞj dsjqðtÞj jxðtÞj dt a
a
Z
a
b
jG1 ðt; tÞqðtÞxðtÞj dt
6
Z
a
¼
b
jG1 ðs; sÞqðsÞxðsÞj ds
a
Z
2
b
jG1 ðt; tÞqðtÞxðtÞj dt
a
Z
b
jG21 ðt; tÞj jqðtÞj dt
6 a
Z
b
jqðtÞjx2 ðtÞ dt:
a
Since Z
b
jqðtÞjx2 ðtÞ dt > 0;
a
so the result follows (where we have used Holder’s inequality).
Theorem 3. Consider the differential equation xð2nÞ þ qðtÞx ¼ 0
ð30Þ
and suppose a solution xðtÞ of (30) satisfies the following boundary value conditions: xðaÞ ¼ x0 ðaÞ ¼ ¼ xðn 1Þ ðaÞ ¼ 0; xðbÞ ¼ x0 ðbÞ ¼ ¼ xðn 1Þ ðbÞ ¼ 0; xðtÞ 6¼ 0, t 2 ða; bÞ, q 2 Cð½a; bÞ. Then Z b 2n 1 2n 1 2 2n 1 jqðtÞjðt aÞ ðb tÞ dt P ð2n 1Þ½ðn 1Þ! ðb aÞ ;
ð31Þ
ð32Þ
a
especially, Z b 2 42n 1 ð2n 1Þ½ðn 1Þ! jqðtÞj dt > : ðb aÞ2n 1 a
ð33Þ
314
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
Proof. By [3], Green’s function Gðt; sÞ of the system (30), (31) is given by 8 n P j n 1 n 1 n 1þj n j 1 ðb tÞðs aÞ > ðt aÞðb sÞ < ð 1Þ ðs tÞ ; t 6 s; j¼0 j ð2n 1Þ! b a b a n P j Gðt; sÞ ¼ n 1 n 1 n 1þj n j 1 ðt aÞðb sÞ ðs aÞðb tÞ > : ð 1Þ ; s 6 t: ðt sÞ j¼0 ð2n 1Þ!
j
b a
b a
Then it is not difficult to prove that max jGðt; sÞj ¼ jGðt; tÞj ¼ jGðs; sÞj ¼ jGðt; sÞjs¼t ¼ G21 ðt; tÞ ¼ G21 ðs; sÞ
a6s6b
2n 1
2
2n 1
and G21 ðt; tÞ ¼ ðt aÞðb tÞ =fð2n 1Þ½ðn 1Þ! ðb aÞ g. Hence (32) follows from Lemma 1 and (33) follows from the inequality ðs aÞðb sÞ < ½ðb aÞ=22
for s 2 ½a; b and 6¼ ða þ bÞ=2:
Remark 4. Let n ¼ 1. We get the result of (2) and (3). The results of (32) and (33) have been obtained in [3], but our method is new and simpler. Theorem 4. Let us consider the following boundary value problem: xð2nÞ þ qðtÞx ¼ 0; xð2iÞ ðaÞ ¼ xð2iÞ ðbÞ ¼ 0;
i ¼ 1; 2; . . . ; n 1:
If xðtÞ is a solution of (34) satisfying xðtÞ 6¼ 0, t 2 ða; bÞ, then Z b 2n jqðtÞj dt > n: ðb aÞ a Proof. First we prove for i ¼ 0; 1; . . . ; n, Z b b a ð2iÞ jx ðtÞj 6 jxð2iþ2Þ ðtÞj dt: 2 a In fact Z t Z t jxð2iÞ ðtÞj ¼ xð2iþ1Þ ðsÞ ds 6 jxð2iþ1Þ ðsÞj ds a
a
and jx
ð2iÞ
Z ðtÞj ¼
b
x t
ð2iþ1Þ
Z ðsÞ ds 6
t
Therefore jxð2iÞ ðtÞj 6
1 2
Z
b
jxð2iþ1Þ ðsÞj ds a
b
jxð2iþ1Þ ðsÞj ds:
ð34Þ
ð35Þ
ð36Þ
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
315
Since xð2iÞ ðaÞ ¼ xð2iÞ ðbÞ ¼ 0, there exists a ci 2 ða; bÞ such that xð2iþ1Þ ðci Þ ¼ 0, and hence Z b Z t Z b ð2iþ1Þ ð2iþ2Þ ð2iþ2Þ jx ðtÞj ¼ x ðsÞ ds 6 jx ðsÞj ds 6 jxð2iþ2Þ ðsÞj ds; ci
ci
a
which yields jxð2iÞ ðtÞj 6
1 2
Z
b
Z
a
b a
Z ðb aÞ b ð2iþ2Þ jxð2iþ2Þ ðsÞjds dt ¼ jx ðsÞj ds: 2 a
Eq. (36) is therefore proved. Let jxðcÞj ¼ maxa 6 t 6 b jxðtÞj. Then by (36), n Z b Z ðb aÞ b 00 b a 0 < jxðcÞj 6 jx ðsÞj ds 6 6 jxð2nÞ ðsÞj ds 2 2 a a n Z b n Z b b a b a ¼ jqðsÞxðsÞj ds 6 jqðsÞj dsjxðcÞj; 2 2 a a which yields (35). Theorem 5. Let us consider the following ðk; n kÞ conjugate boundary value problem: ð 1Þ
n k ðnÞ
x
xðiÞ ðaÞ ¼ 0; ðjÞ
x ðbÞ ¼ 0;
¼ qðtÞx;
ð37Þ
a < t < b;
0 6 i 6 k 1;
ð38Þ
0 6 j 6 n k 1:
If xðtÞ is a solution of (37) and (38) satisfying xðtÞ ¼ 0, t 2 ða; bÞ, then Z b uðtÞjqðtÞj dt > 1;
ð39Þ
a
where uðtÞ ¼ maxfu1 ðtÞ; u2 ðtÞg and n k
u1 ðtÞ ¼ ½ðt aÞðb tÞ
½ðb aÞ þ ðt aÞ
k
u2 ðtÞ ¼ ½ðb tÞðt aÞ ½ðb aÞ þ ðb tÞ especially, we have Z b jqðtÞj dt > maxfA; Bg;
=½ðn kÞ!ðk 1Þ!ðb aÞ
n k 1
n k
;
k
=½ðn k 1Þ!k!ðb aÞ ;
ð40Þ
a
where A ¼ ðn kÞ!ðk 1Þ!22n 3kþ1 =ðb aÞ
k 1
2n 1
B ¼ ðn k 1Þ!k!23k nþ1 =ðb aÞn 1 :
;
316
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
Proof. Let Gðt; sÞ be Green’s function of the boundary value problem xðnÞ ¼ 0, xðiÞ ðaÞ ¼ 0, 0 6 i 6 k 1, xðjÞ ðbÞ ¼ 0, 0 6 j 6 n k 1. Then by [7], i 8P hP k 1 j n kþi 1 t a i ðt aÞj ða sÞn j 1 b t k < k 1 ; s 6 t; j¼0 i¼0 i b a b a j!ðn j 1Þ! hP i Gðt; sÞ ¼
j n j 1 P n k 1 j kþi 1 b t i ðt bÞ ðb sÞ t a k : n k 1 ; t 6 s: j¼0 i¼0 i b a b a j!ðn j 1Þ! Hence for a 6 s 6 t 6 b, ðt aÞ=ðb aÞ 6 1, b t 6 b s, t a 6 b a, we have " # k 1 j k 1 X n k þ i 1 ðb aÞj ðs aÞn j 1 ðb sÞn k X jGðt; sÞj 6 n k i j!ðn j 1Þ!ðb aÞ j¼0 i¼0 j n j 1 n k k 1 X n j 1 ðb aÞ ðs aÞ ðb sÞ ¼ k j 1 j!ðn j 1Þ!ðb aÞn k j¼0 ¼
ðs aÞn k ðb sÞn k ðk 1Þ!ðn kÞ!ðb aÞ
k 1
n k
½ðb aÞ þ ðs aÞ
¼ u1 ðsÞ;
for a 6 t 6 s 6 b, ðb tÞ=ðb aÞ 6 1, t a 6 s a, b t 6 b a, we have " # n k 1 X k þ i 1 ðb aÞj ðb sÞn j 1 s a k X n k 1 j jGðt; sÞj 6 i b a j!ðn j 1Þ! i¼0 j¼0 n k 1 X n j 1 ðb aÞj ðb sÞn j 1 s a k ¼ n k j 1 b a j!ðn j 1Þ! j¼0 ¼
ðs aÞk ðb sÞk
n k 1
k!ðn k 1Þ!ðb aÞ
k
½ðb aÞ þ ðb sÞ
¼ u2 ðsÞ:
Let xðtÞ be a solution of (37) and (38) satisfying xðtÞ 6¼ 0, t 2 ða; bÞ, and let jxðcÞj ¼ maxa 6 t 6 b jxðtÞj. Then Z b Gðc; sÞqðsÞxðsÞ ds 0 < jxðcÞj ¼ a Z b Z b 6 jGðc; sÞqðsÞj dsjxðcÞj 6 uðsÞjqðsÞj ds jxðcÞj; a
a
which yields (39). Since ðs aÞðb sÞ ¼
b a 2
2
2 2 aþb b a s < 2 2
aþb ; 2 ðb aÞ þ ðt aÞ 6 2ðb aÞ; for s 6¼
we obtain (40).
ðb aÞ þ ðb tÞ 6 2ðb aÞ;
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
317
References [1] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964. [2] W.T. Reid, A generalized Liapunov inequality, J. Differential Equations 13 (1973) 182–196. [3] K.M. Das, A.S. Vatsala, Green’s function for n–n boundary value problem and an analogue of Hartman’s result, J. Math. Anal. Appl. 51 (1975) 670–677. [4] B.G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations, J. Math. Anal. Appl. 195 (1995) 527–536. [5] N. Parhi, S. Panigrahi, On Liapunov-type inequality for third-order differential equations, J. Math. Anal. Appl. 233 (1999) 445–464. [6] P.R. Beesack, On Green’s function of an N-point boundary value problem, Pacific J. Math. 12 (1962) 801–812. [7] G.G. Gustafson, A Green’s function convergence principle, with applications to computation and norm estimates, Rocky Mountain J. Math. 6 (1976) 457–492.
On Liapunov-type inequality for certain higher-order differential equations Xiaojing Yang Department of Mathematics, Tsinghua University, Beijing 100084, PR China
Abstract In this paper, we generalize the well-known Liapunov-type inequality for a class of third-order differential equations without damping force to 2n þ 1 order differential equations. Ó 2002 Published by Elsevier Science Inc. Keywords: Liapunov’s inequalities; Green’s functions; Higher-order equations
1. Introduction It is well known that if xðtÞ is a solution of x00 þ qðtÞx ¼ 0 satisfying xðaÞ ¼ xðbÞ ¼ 0 ða < bÞ and xðtÞ 6¼ 0 for t 2 ða; bÞ, then Z b jqðtÞj dt > 4=ðb aÞ:
ð1Þ
ð2Þ
a
Hartman [1] generalized (2) as follows: Z b ðt aÞðb tÞqþ ðtÞ dt > b a; a
where qþ ðtÞ ¼ maxfqðtÞ; 0g.
E-mail address: [email protected] (X. Yang). 0096-3003/02/$ - see front matter Ó 2002 Published by Elsevier Science Inc. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 2 8 5 - 5
ð3Þ
308
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
Generalizations or improvements of (2) have been obtained by many authors, see e.g. [2–4]. Recently Parhi and Panigrahi [5] established an inequality similar to (2) for third-order differential equation of the form x000 þ qðtÞx ¼ 0;
ð4Þ
where q 2 Cð½0; 1Þ; RÞ. Their results are: Theorem A. If there exists a d 2 ða; bÞ such that x00 ðdÞ ¼ 0, then Z b 2 jqðtÞj dt > 4=ðb aÞ :
ð5Þ
a
Theorem B. If x00 ðtÞ 6¼ 0, t 2 ða; bÞ, xðtÞ has three consecutive zeros a < b < a0 , then Z a0 2 jqðtÞj dt > 4=ða0 aÞ : ð6Þ a
In this paper, we generalize the inequalities of (2), (3), (5) and (6) to certain higher-order differential equations.
2. Main results Theorem 1. Let n 2 N , qðtÞ 2 Cð½a; bÞ. If there exists a d 2 ða; bÞ such that xð2nÞ ðdÞ ¼ 0; where xðtÞ is a solution of the following differential equation xð2nþ1Þ ðtÞ þ gðtÞxðtÞ ¼ 0
ð7Þ
satisfying xðiÞ ðaÞ ¼ xðiÞ ðbÞ ¼ 0; i ¼ 0; 1; . . . ; n 1;
xðtÞ 6¼ 0; t 2 ða; bÞ;
ð8Þ
then Z
b 2n
jqðtÞj dt > n!2ðnþ1Þ =ðb aÞ :
ð9Þ
a
Proof. Since if xðtÞ is a solution of (7), then xðtÞ is also a solution of (7), we can assume xðtÞ > 0, t 2 ða; bÞ. Since xðaÞ ¼ xðbÞ ¼ 0, there exists a c 2 ða; bÞ such that x0 ðcÞ ¼ 0 and xðcÞ ¼ maxa 6 t 6 b xðtÞ. Thus Z c Z c 0 < xðcÞ ¼ x0 ðtÞ dt 6 jx0 ðtÞj dt: ð10Þ a
a
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
309
Since x0 ðaÞ ¼ x0 ðcÞ ¼ 0; there is a t21 2 ða; cÞ such that x00 ðt21 Þ ¼ 0 and from x0 ðbÞ ¼ x0 ðcÞ ¼ 0; there is a t22 2 ðc; bÞ such that x00 ðt22 Þ ¼ 0. In this way, we see there exist ftjk gjk¼1 ða; bÞ such that xðjÞ ðtjk Þ ¼ 0; k ¼ 1; . . . ; j, j ¼ 1; 2; . . . ; n, n j jþ1 1 k and 0 < tjþ1 < tj1 < c < tjj < tjþ1 < b, j ¼ 1; . . . ; n. Hence there exist ftjþn gk¼1 k ða; bÞ such that xðnþjÞ ðtnþj Þ ¼ 0, k ¼ 1; . . . ; n j, j ¼ 1; . . . ; n 1. 0 From x ðaÞ ¼ 0, we have Z t x0 ðtÞ ¼ x00 ðsÞ ds: a
Hence jx0 ðtÞj 6
Z
t
jx00 ðsÞj ds;
t 2 ½a; c:
a
By (10), Z
c
jx0 ðtÞj dt 6
xðcÞ 6 a
Z cZ a
jx00 ðsÞj ds dt:
t
By using the equality Z cZ t Z cZ c f ðt; sÞ ds dt ¼ f ðt; sÞ dt ds; a
a
ð11Þ
a
a
ð12Þ
s
we obtain Z cZ
c 00
jx ðsÞj dt ds ¼
xðcÞ 6 a
s
Z
c
ðc sÞjx00 ðsÞj ds:
ð13Þ
a
Since for 1 6 k 6 n 1, Z t Z t ðkÞ ðkþ1Þ jx ðtÞj ¼ x ðsÞ ds 6 jxðkþ1Þ ðsÞj ds; a
a
it is easy to obtain Z
c
xðcÞ 6 a
ðc tÞn 1 ðnÞ jx ðtÞj dt: ðn 1Þ!
ð14Þ
k 1 From above analysis, there exist ftnþj gn j k¼1 , j ¼ 1; . . . ; n 1; and tn 2 ða; cÞ, n k tn 2 ðc; bÞ, such that xðtnþj Þ ¼ 0, k ¼ 1; . . . ; n j. Therefore Z t Z c jxðnÞ ðtÞj 6 jxðnþ1Þ ðsÞj ds 6 jxðnþ1Þ ðsÞj ds; t 2 ½a; c; tn1
a
310
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
and jxðnþ1Þ ðtÞj 6
Z
t
jxðnþ2Þ ðsÞj ds 6 1 tnþ1
Z
b
jxðnþ2Þ ðsÞj ds 6
a
6 6
Z
b
Z
Z
b
Z
b
Z
a
b
jxðnþ3Þ ðsÞj ds
a
b
jxð2nÞ ðsÞj ds dt1 dtm : a a |fflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflaffl}
ð15Þ
n 1
By assumption, there exists a d 2 ða; bÞ such that xð2nÞ ðdÞ ¼ 0. This gives Z t Z b jxð2nþ1Þ ðsÞj ds 6 jxð2nþ1Þ ðsÞj ds jxð2nÞ ðtÞj 6 d a Z b Z b jqðsÞxðsÞj ds 6 jqðsÞj ds xðcÞ: ð16Þ ¼ a
a
Combining (14)–(16), we obtain " #Z Z b ðb aÞn 1 ðc aÞ c n 1 ðc tÞ dt jqðtÞj dt xðcÞ 0 < xðcÞ 6 ðn 1Þ! a a Z b xðcÞ ðnþ1Þ ðn 1Þ ðc aÞ 6 ðb aÞ jqðtÞj dt; n! a which means Z b jqðtÞj dt P a
n! ðb aÞ
n 1
ðc aÞ
nþ1
ð17Þ
:
Similarly, we can prove that Z b n! jqðtÞj dt P : n 1 nþ1 ðb aÞ ðb cÞ a
ð18Þ
Since the function f ðuÞ ¼ u ðnþ1Þ is convex (f 00 ðuÞ > 0 for u > 0), we have 1 ðb cÞ
þ nþ1
1 ðc aÞ
nþ1
>
2
¼ b a nþ1 2
2nþ2 ðb aÞ
nþ1
this, together with (17) and (18), yields Z b n!2nþ1 jqðtÞj dt > : 2n ðb aÞ a Remark 1. Let n ¼ 1. We get the result of Theorem A [5], moreover, from (17) and (18), we see c cannot be very close to a as well as to b, which is also a generalization of Theorem 3 of [5].
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
311 N
Corollary 1. If xðtÞ is a solution of (7), which has N zeros ftk gk¼1 in the interval ½a; b, a 6 t1 < t2 < < tN 6 b, and for each k ¼ 1; . . . ; N 1, there exists a dk 2 ðtk ; tkþ1 Þ such that xð2nÞ ðdk Þ ¼ 0, then 2n Z b ðb aÞ 2nþ1 < jqðtÞj dt: ð19Þ ðN 1Þ n!2nþ1 a Proof. From Theorem 1, for k ¼ 1; . . . ; N 1, Z tkþ1 n!2nþ1 jqðtÞj dt > : 2n ðtkþ1 tk Þ tk Hence Z
b
jqðtÞj P
a
N 1 Z X k¼1
tkþ1
jqðtÞj dt > n!2nþ1
tk
N 1 X k¼1
1 ðtkþ1 tk Þ
2n
:
ð20Þ
Since f ðuÞ ¼ u 2n is convex for u > 0, we have for xk ¼ tkþ1 tk > 0, k ¼ 1; . . . ; N 1, ! N 1 N 1 X X f ðxk Þ > ðN 1Þf xk =ðN 1Þ ; k¼1
k¼1
that is N 1 X
1
k¼1
ðtkþ1 tk Þ2n
>
ðN 1Þ
2nþ1
ðtN t1 Þ2n
Hence Eqs. (20) and (21) give (19).
2nþ1
P
ðN 1Þ
ðb aÞ2n
:
ð21Þ
Remark 2. Let n ¼ 1. Then the result of (19) is sharper than the result of Theorem 6 of [5] for N P 3. Theorem 2. Let n 2 N , n P 2, qðtÞ 2 Cð½a; bÞ. If the differential equation xðnÞ þ qðtÞx ¼ 0
ð22Þ
has a solution xðtÞ satisfying the boundary value problem xðaÞ ¼ xðt2 Þ ¼ ¼ xðtn 1 Þ ¼ xðbÞ ¼ 0;
ð23Þ
where a ¼ t1 < t2 < < tn 1 < tn ¼ b and xðtÞ 6¼ 0, t 2 ðtk ; tkþ1 Þ, k ¼ 1; . . . ; n 1, then Z b ðn 2Þ!nn 1 jqðtÞj dt > : ð24Þ ðn 1Þn 2 ðb aÞn 1 a
312
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
Proof. Let Gðt; sÞ be Green’s function for the following boundary value problem xðnÞ ðtÞ ¼ 0, xðt1 Þ ¼ xðt2 Þ ¼ ¼ xðtn Þ ¼ 0. Then Z b xðtÞ ¼ Gðt; sÞqðsÞxðsÞ ds: ð25Þ a
Beesack [6] proved that jGðt; sÞj 6
n Y ðt tk Þ=½ðn 1Þ!ðb aÞ
ð26Þ
k¼1
and for t 2 ½a; b, jGðt; sÞj 6
n Y
n 1
jt tk j=½ðn 1Þ!ðb aÞ 6
k¼1 n 2
¼
ðn 1Þ ðb aÞ ðn 2Þ!nn 1
ðn 1Þ ðb aÞ ðn 1Þ!nn 1
n 1
n 1
:
Let jxðcÞj ¼ maxa 6 t 6 b jxðtÞj. Then from (25) and (26), we obtain Z b 0 < jxðcÞj 6 jGðt; sÞj jqðsÞj jxðsÞj ds a
Z < a
b
ðn 1Þn 2 ðb aÞn 1 jqðsÞj ds jxðcÞj ; ðn 2Þ!nn 1
which yields (24). Remark 3. Let n ¼ 3. We have Z b 9 jgðtÞj dt > ; 2 2ðb aÞ a which is sharper than the result of [5]. Lemma 1. Let us consider the differential equation (22). If xðtÞ is a nontrivial solution of (22) satisfying some two-point boundary value problem li ðxðaÞÞ ¼ 0;
l~i ðxðbÞÞ ¼ 0;
i ¼ 1; . . . ; m;
ð27Þ
and Green’s function Gðt; sÞ for the boundary value problem xðnÞ ¼ 0, li ðaÞ ¼ l~i ðbÞ ¼ 0, i ¼ 1; . . . ; m, satisfies jGðt; sÞj 6 jG1 ðt; tÞG1 ðs; sÞj;
t; s 2 ½a; b:
ð28Þ
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
313
Moreover, qðtÞ 6 0; xðtÞ 6¼ 0, t 2 ða; bÞ, then Z b jG21 ðt; tÞqðtÞj dt P 1:
ð29Þ
a
Proof. Multiplying both sides of (25) by jqðtÞjxðtÞ, then integrating over ½a; b we have Z b Z bZ b 2 jqðtÞjx ðtÞ dt 6 jGðt; sÞj jqðsÞj jxðsÞj dsjqðtÞj jxðtÞj dt a
a
Z
a
b
jG1 ðt; tÞqðtÞxðtÞj dt
6
Z
a
¼
b
jG1 ðs; sÞqðsÞxðsÞj ds
a
Z
2
b
jG1 ðt; tÞqðtÞxðtÞj dt
a
Z
b
jG21 ðt; tÞj jqðtÞj dt
6 a
Z
b
jqðtÞjx2 ðtÞ dt:
a
Since Z
b
jqðtÞjx2 ðtÞ dt > 0;
a
so the result follows (where we have used Holder’s inequality).
Theorem 3. Consider the differential equation xð2nÞ þ qðtÞx ¼ 0
ð30Þ
and suppose a solution xðtÞ of (30) satisfies the following boundary value conditions: xðaÞ ¼ x0 ðaÞ ¼ ¼ xðn 1Þ ðaÞ ¼ 0; xðbÞ ¼ x0 ðbÞ ¼ ¼ xðn 1Þ ðbÞ ¼ 0; xðtÞ 6¼ 0, t 2 ða; bÞ, q 2 Cð½a; bÞ. Then Z b 2n 1 2n 1 2 2n 1 jqðtÞjðt aÞ ðb tÞ dt P ð2n 1Þ½ðn 1Þ! ðb aÞ ;
ð31Þ
ð32Þ
a
especially, Z b 2 42n 1 ð2n 1Þ½ðn 1Þ! jqðtÞj dt > : ðb aÞ2n 1 a
ð33Þ
314
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
Proof. By [3], Green’s function Gðt; sÞ of the system (30), (31) is given by 8 n P j n 1 n 1 n 1þj n j 1 ðb tÞðs aÞ > ðt aÞðb sÞ < ð 1Þ ðs tÞ ; t 6 s; j¼0 j ð2n 1Þ! b a b a n P j Gðt; sÞ ¼ n 1 n 1 n 1þj n j 1 ðt aÞðb sÞ ðs aÞðb tÞ > : ð 1Þ ; s 6 t: ðt sÞ j¼0 ð2n 1Þ!
j
b a
b a
Then it is not difficult to prove that max jGðt; sÞj ¼ jGðt; tÞj ¼ jGðs; sÞj ¼ jGðt; sÞjs¼t ¼ G21 ðt; tÞ ¼ G21 ðs; sÞ
a6s6b
2n 1
2
2n 1
and G21 ðt; tÞ ¼ ðt aÞðb tÞ =fð2n 1Þ½ðn 1Þ! ðb aÞ g. Hence (32) follows from Lemma 1 and (33) follows from the inequality ðs aÞðb sÞ < ½ðb aÞ=22
for s 2 ½a; b and 6¼ ða þ bÞ=2:
Remark 4. Let n ¼ 1. We get the result of (2) and (3). The results of (32) and (33) have been obtained in [3], but our method is new and simpler. Theorem 4. Let us consider the following boundary value problem: xð2nÞ þ qðtÞx ¼ 0; xð2iÞ ðaÞ ¼ xð2iÞ ðbÞ ¼ 0;
i ¼ 1; 2; . . . ; n 1:
If xðtÞ is a solution of (34) satisfying xðtÞ 6¼ 0, t 2 ða; bÞ, then Z b 2n jqðtÞj dt > n: ðb aÞ a Proof. First we prove for i ¼ 0; 1; . . . ; n, Z b b a ð2iÞ jx ðtÞj 6 jxð2iþ2Þ ðtÞj dt: 2 a In fact Z t Z t jxð2iÞ ðtÞj ¼ xð2iþ1Þ ðsÞ ds 6 jxð2iþ1Þ ðsÞj ds a
a
and jx
ð2iÞ
Z ðtÞj ¼
b
x t
ð2iþ1Þ
Z ðsÞ ds 6
t
Therefore jxð2iÞ ðtÞj 6
1 2
Z
b
jxð2iþ1Þ ðsÞj ds a
b
jxð2iþ1Þ ðsÞj ds:
ð34Þ
ð35Þ
ð36Þ
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
315
Since xð2iÞ ðaÞ ¼ xð2iÞ ðbÞ ¼ 0, there exists a ci 2 ða; bÞ such that xð2iþ1Þ ðci Þ ¼ 0, and hence Z b Z t Z b ð2iþ1Þ ð2iþ2Þ ð2iþ2Þ jx ðtÞj ¼ x ðsÞ ds 6 jx ðsÞj ds 6 jxð2iþ2Þ ðsÞj ds; ci
ci
a
which yields jxð2iÞ ðtÞj 6
1 2
Z
b
Z
a
b a
Z ðb aÞ b ð2iþ2Þ jxð2iþ2Þ ðsÞjds dt ¼ jx ðsÞj ds: 2 a
Eq. (36) is therefore proved. Let jxðcÞj ¼ maxa 6 t 6 b jxðtÞj. Then by (36), n Z b Z ðb aÞ b 00 b a 0 < jxðcÞj 6 jx ðsÞj ds 6 6 jxð2nÞ ðsÞj ds 2 2 a a n Z b n Z b b a b a ¼ jqðsÞxðsÞj ds 6 jqðsÞj dsjxðcÞj; 2 2 a a which yields (35). Theorem 5. Let us consider the following ðk; n kÞ conjugate boundary value problem: ð 1Þ
n k ðnÞ
x
xðiÞ ðaÞ ¼ 0; ðjÞ
x ðbÞ ¼ 0;
¼ qðtÞx;
ð37Þ
a < t < b;
0 6 i 6 k 1;
ð38Þ
0 6 j 6 n k 1:
If xðtÞ is a solution of (37) and (38) satisfying xðtÞ ¼ 0, t 2 ða; bÞ, then Z b uðtÞjqðtÞj dt > 1;
ð39Þ
a
where uðtÞ ¼ maxfu1 ðtÞ; u2 ðtÞg and n k
u1 ðtÞ ¼ ½ðt aÞðb tÞ
½ðb aÞ þ ðt aÞ
k
u2 ðtÞ ¼ ½ðb tÞðt aÞ ½ðb aÞ þ ðb tÞ especially, we have Z b jqðtÞj dt > maxfA; Bg;
=½ðn kÞ!ðk 1Þ!ðb aÞ
n k 1
n k
;
k
=½ðn k 1Þ!k!ðb aÞ ;
ð40Þ
a
where A ¼ ðn kÞ!ðk 1Þ!22n 3kþ1 =ðb aÞ
k 1
2n 1
B ¼ ðn k 1Þ!k!23k nþ1 =ðb aÞn 1 :
;
316
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
Proof. Let Gðt; sÞ be Green’s function of the boundary value problem xðnÞ ¼ 0, xðiÞ ðaÞ ¼ 0, 0 6 i 6 k 1, xðjÞ ðbÞ ¼ 0, 0 6 j 6 n k 1. Then by [7], i 8P hP k 1 j n kþi 1 t a i ðt aÞj ða sÞn j 1 b t k < k 1 ; s 6 t; j¼0 i¼0 i b a b a j!ðn j 1Þ! hP i Gðt; sÞ ¼
j n j 1 P n k 1 j kþi 1 b t i ðt bÞ ðb sÞ t a k : n k 1 ; t 6 s: j¼0 i¼0 i b a b a j!ðn j 1Þ! Hence for a 6 s 6 t 6 b, ðt aÞ=ðb aÞ 6 1, b t 6 b s, t a 6 b a, we have " # k 1 j k 1 X n k þ i 1 ðb aÞj ðs aÞn j 1 ðb sÞn k X jGðt; sÞj 6 n k i j!ðn j 1Þ!ðb aÞ j¼0 i¼0 j n j 1 n k k 1 X n j 1 ðb aÞ ðs aÞ ðb sÞ ¼ k j 1 j!ðn j 1Þ!ðb aÞn k j¼0 ¼
ðs aÞn k ðb sÞn k ðk 1Þ!ðn kÞ!ðb aÞ
k 1
n k
½ðb aÞ þ ðs aÞ
¼ u1 ðsÞ;
for a 6 t 6 s 6 b, ðb tÞ=ðb aÞ 6 1, t a 6 s a, b t 6 b a, we have " # n k 1 X k þ i 1 ðb aÞj ðb sÞn j 1 s a k X n k 1 j jGðt; sÞj 6 i b a j!ðn j 1Þ! i¼0 j¼0 n k 1 X n j 1 ðb aÞj ðb sÞn j 1 s a k ¼ n k j 1 b a j!ðn j 1Þ! j¼0 ¼
ðs aÞk ðb sÞk
n k 1
k!ðn k 1Þ!ðb aÞ
k
½ðb aÞ þ ðb sÞ
¼ u2 ðsÞ:
Let xðtÞ be a solution of (37) and (38) satisfying xðtÞ 6¼ 0, t 2 ða; bÞ, and let jxðcÞj ¼ maxa 6 t 6 b jxðtÞj. Then Z b Gðc; sÞqðsÞxðsÞ ds 0 < jxðcÞj ¼ a Z b Z b 6 jGðc; sÞqðsÞj dsjxðcÞj 6 uðsÞjqðsÞj ds jxðcÞj; a
a
which yields (39). Since ðs aÞðb sÞ ¼
b a 2
2
2 2 aþb b a s < 2 2
aþb ; 2 ðb aÞ þ ðt aÞ 6 2ðb aÞ; for s 6¼
we obtain (40).
ðb aÞ þ ðb tÞ 6 2ðb aÞ;
X. Yang / Appl. Math. Comput. 134 (2003) 307–317
317
References [1] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964. [2] W.T. Reid, A generalized Liapunov inequality, J. Differential Equations 13 (1973) 182–196. [3] K.M. Das, A.S. Vatsala, Green’s function for n–n boundary value problem and an analogue of Hartman’s result, J. Math. Anal. Appl. 51 (1975) 670–677. [4] B.G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations, J. Math. Anal. Appl. 195 (1995) 527–536. [5] N. Parhi, S. Panigrahi, On Liapunov-type inequality for third-order differential equations, J. Math. Anal. Appl. 233 (1999) 445–464. [6] P.R. Beesack, On Green’s function of an N-point boundary value problem, Pacific J. Math. 12 (1962) 801–812. [7] G.G. Gustafson, A Green’s function convergence principle, with applications to computation and norm estimates, Rocky Mountain J. Math. 6 (1976) 457–492.