Lyapunov-type inequality for quasilinear systems

Applied Mathematics and Computation 219 (2012) 1670–1673

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Lyapunov-type inequality for quasilinear systems Xiaojing Yang a, Yong-In Kim b,⇑, Kueiming Lo c a b c

Department of Mathematics, Tsinghua University, Beijing 100084, China Department of Mathematics, University of Ulsan, Ulsan 680-749, Republic of Korea School of Software, Tsinghua University, Key Laboratory for Information System Security, Ministry of Education of China, Beijing 100084, China

a r t i c l e

i n f o

a b s t r a c t Some new version of the well-known Lyapunov-type inequality for a class of quasilinear systems is given. The results of this paper generalize some previous results on this topic. Ó 2012 Elsevier Inc. All rights reserved.

Keywords: Lyapunov-type inequality Quasilinear system

1. Introduction It is well-known that the Lyapunov inequality for the second-order linear differential equation

x00 þ qðtÞx ¼ 0

ð1Þ

states that if qðtÞ P 0 is continuous and Eq. (1) has a nonzero solution xðtÞ satisfying the boundary condition:

xðaÞ ¼ xðbÞ ¼ 0;

xðtÞ X 0; t 2 ða; bÞ

then

Z

b

qðtÞ dt >

a

4 : ba

This result has found many applications in the study of various properties of solutions of differential equations such as oscillation theory, disconjugacy and eigenvalue problems. There have been many proofs and generalizations of the Lyapunov inequality. For example, see the papers [1–10] and the references therein. But so far, there has been only a few results obtained for differential systems. Recently, De Nápoli and Pinasco [6] have obtained the following results: Theorem A. Consider the following ðp; qÞ-quasilinear system:

ð/p ðu0 ÞÞ0 þ f1 ðtÞjuja2 ujv jb ¼ 0; ð/q ðv 0 ÞÞ0 þ f2 ðtÞjuja jv jb2 v ¼ 0;

ð2Þ

where 1 < p; q < þ1; /p ðuÞ ¼ jujp2 u; f1 , f2 are real nonnegative continuous functions and a, b are non-negative numbers satisfying

a b p

þ

q

¼ 1:

⇑ Corresponding author. E-mail addresses: [email protected] (X. Yang), [email protected] (Y.-I. Kim), [email protected] (K. Lo). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.08.007

ð3Þ

X. Yang et al. / Applied Mathematics and Computation 219 (2012) 1670–1673

1671

If fi ðtÞ > 0 for t 2 ½a; b and i ¼ 1; 2 and Eq. (1) has a nontrivial solution ðu; v Þ satisfying the boundary condition uðaÞ ¼ uðbÞ ¼ v ðaÞ ¼ v ðbÞ ¼ 0, then the following inequality holds: aþb

2

Z

aþb1

6 ðb  aÞ

b

f1 ðtÞ dt

!ap Z

a

!bq

b

f2 ðtÞ dt

ð4Þ

:

a

More recently, Çakmak and Tiryaki [1] have considered the following 2-dimensional system:

ðr 1 ðtÞ/p ðu0 ÞÞ0 þ f1 ðtÞjuja2 ujv jb ¼ 0;

ð5Þ

ðr 2 ðtÞ/q ðv 0 ÞÞ0 þ f2 ðtÞjujh jv jc2 v ¼ 0; where (i) r1 ; r2 ; f1 ; f2 are real valued continuous functions such that r1 ðtÞ > 0 and r 2 ðtÞ > 0 for all t 2 R. (ii) the exponents satisfy 1 < p; q < þ1 and the positive parameters a; b; h and c satisfy ap þ bq ¼ 1 and ph þ qc ¼ 1. They have obtained the following result:

Theorem B. If system (5) has a real nontrivial solution ðuðtÞ; v ðtÞÞ such that uðaÞ ¼ uðbÞ ¼ v ðaÞ ¼ v ðbÞ ¼ 0 where a; b 2 R with a < b be consecutive zeros, then the following inequality holds: hþb

2

Z

b

1 1p

ðr 1 ðtÞÞ

6

dt

!hðp1Þ Z p

a

b

1 1q

ðr 2 ðtÞÞ

dt

!bðq1Þ Z q

a

a

b

f1þ ðtÞ dt

!ph Z a

!bq

b

f2þ ðtÞ dt

;

ð6Þ

where fiþ ðtÞ ¼ maxffi ðtÞ; 0g for i ¼ 1; 2. In this paper, we generalize the above results to a class of 3-dimensional quasilinear differential systems. 2. Main results Let us consider the following 3-dimensional quasilinear system:

ðr 1 ðtÞ/p1 ðx0 ÞÞ0 þ f1 ðtÞ/q1;1 ðxÞwq1;2 ðyÞwq1;3 ðzÞ ¼ 0; ðr 2 ðtÞ/p2 ðy0 ÞÞ0 þ f2 ðtÞwq2;1 ðxÞ/q2;2 ðyÞwq2;3 ðzÞ ¼ 0; 0

ð7Þ

0

ðr 3 ðtÞ/p3 ðz ÞÞ þ f3 ðtÞwq3;1 ðxÞwq3;2 ðyÞ/q3;3 ðzÞ ¼ 0; where 1 < pk < þ1 for k ¼ 1; 2; 3; qi;j are nonnegative constants for i; j ¼ 1; 2; 3; /p ðuÞ ¼ jujp2 u; wq ðuÞ ¼ jujq1 u for q > 0 with w0 ðuÞ ¼ signðuÞ ¼ 1 for q ¼ 0; rk 2 C 1 ð½a; b; ð0; þ1ÞÞ for k ¼ 1; 2; 3 and fk ðtÞ 2 C½a; b for k ¼ 1; 2; 3. Definition 1. We call a real vector ða1 ; a2 ; a3 Þ positive if ai P 0 for i ¼ 1; 2; 3 and a1 þ a2 þ a3 > 0. The main result of this paper is the following theorem. Theorem 1. Let a < b and assume that there exist a positive solution ðe1 ; e2 ; e3 Þ of the following linear homogeneous system:

ðp1  q1;1 Þe1  q2;1 e2  q3;1 e3 ¼ 0;  q1;2 e1 þ ðp2  q2;2 Þe2  q3;2 e3 ¼ 0;

ð8Þ

 q1;3 e1  q2;3 e2 þ ðp3  q3;3 Þe3 ¼ 0: Suppose that there exists a nonzero solution ðxðtÞ; yðtÞ; zðtÞÞ of (7) satisfying xðaÞ ¼ xðbÞ ¼ yðaÞ ¼ yðbÞ ¼ zðaÞ ¼ zðbÞ ¼ 0 and ðxðtÞ; yðtÞ; zðtÞÞ X ð0; 0; 0Þ. Then we have

Z 3 Y k¼1

a

!e k

b

fkþ ðtÞ dt

>2

Q

Z 3 Y k¼1

b

1 1pk

ðr k ðtÞÞ

!ð1pk Þek

dt

;

ð9Þ

a

where fkþ ðtÞ ¼ maxffk ðtÞ; 0g for k ¼ 1; 2; 3 and Q ¼

P3

j¼1 pj ej .

Corollary 1. Assume that 3 X qj;k ¼ pk ; j¼1

k ¼ 1; 2; 3:

ð10Þ

1672

X. Yang et al. / Applied Mathematics and Computation 219 (2012) 1670–1673

Suppose that there exists a nonzero solution ðxðtÞ; yðtÞ; zðtÞÞ of (7) which satisfies xðaÞ ¼ xðbÞ ¼ yðaÞ ¼ yðbÞ ¼ zðaÞ ¼ zðbÞ ¼ 0 and ðxðtÞ; yðtÞ; zðtÞÞ X ð0; 0; 0Þ. Then we have 3 Z Y

b

fkþ ðtÞ dt

a

k¼1

>2

P

Z 3 Y

b

1 1pk

ðr k ðtÞÞ

!1pk

dt

ð11Þ

;

a

k¼1

where fkþ ðtÞ ¼ maxffk ðtÞ; 0g for k ¼ 1; 2; 3 and P ¼

P3

j¼1 pj .

Corollary 2. Assume that det A3 ¼ 0, where

0

p1  q1;1

B A3 ¼ @ q1;2 q1;3

q2;1

q3;1

p2  q2;2 q2;3

1

q3;2 C A: p2  q3;3

Suppose that (I) rank A3 ¼ 1 and pk  qk;k P 0 for k ¼ 1; 2; 3 and qi;j P 0 for i; j ¼ 1; 2; 3; or that (II) rank A3 ¼ 2 and at least one of the following vectors is positive:

a1 ¼ ðq2;1 q3;2 þ q3;1 ðp2  q2;2; Þ; q1;2 q3;1 þ q3;2 ðp1  q1;1 Þ; ðp1  q1;1 Þðp2  q2;2 Þ  q1;2 q2;1 Þ; a2 ¼ ððp2  q2;2 Þðp3  q3;3 Þ  q2;3 q3;2 ; ðp3  q3;3 Þq1;2 þ q1;3 q3;2 ; q1;2 q2;3 þ q1;3 ðp2  q2;2 ÞÞ; a3 ¼ ðq3;1 q2;3 þ q2;1 ðp3  q3;3 Þ; ðp1  q1;1 Þðp3  q3;3 Þ  q1;3 q3;1 ; q2;1 q1;3 þ q2;3 ðp1  q1;1 ÞÞ:

ð12Þ

Then Eq. (8) has a positive solution ðe1 ; e2 ; e3 Þ such that inequality(9) holds. 3. Proof of the main results

Proof of Theorem 1. Consider the first equation of (7) and assume that jxðcÞj ¼ maxa6t6b jxðtÞj for some c 2 ða; bÞ. Then from xðtÞ X 0, we see that jxðcÞj > 0 and x0 ðcÞ ¼ 0. Then from xðaÞ ¼ 0 and using Hölder’s inequality, we obtain

Z c  Z   jxðcÞj ¼  x0 ðtÞ dt 6 a

¼

c

jx0 ðtÞj dt ¼

a

Z

c

1 1p1

ðr 1 ðtÞÞ

dt

Z

c

p1

ðr 1 ðtÞÞ

1

1

ðr 1 ðtÞÞp1 jx0 ðtÞj dt 6

a

 10 Z p

c

1

a

Z a

r1 ðtÞjx0 ðtÞjp1 dt

p0

c

ðr 1 ðtÞÞ

p1 1

dt

p10 Z 1

c

r1 ðtÞjx0 ðtÞjp1 dt

p1

1

a

p1

1

;

ð13Þ

r 1 ðtÞjx0 ðtÞjp1 dt:

ð14Þ

a

1 where p01 ¼ p p1 . From (13), we obtain 1

p1

jxðcÞj

Z

6

c

1

ðr 1 ðtÞÞ1p1 dt

p1 1 Z

a

c

a

Multiplying the first equation of (7) by xðtÞ and integrating over ½a; c and using integration by parts, we obtain



Z

c a

Z c Z c ðr 1 ðtÞ/p1 ðx0 ðtÞÞÞ0 xðtÞ dt ¼ r 1 ðtÞ/p1 ðx0 ðtÞÞxðtÞjca þ r1 ðtÞjxðtÞjp1 dt ¼ r 1 ðtÞjx0 ðtÞjp1 dt a a Z c Z c ¼ f1 ðtÞ/q1;1 ðxðtÞÞxðtÞwq1;2 ðyðtÞÞwq1;3 ðzðtÞÞ dt 6 f1þ ðtÞjxðtÞjq1;1 jyðtÞjq1;2 jzðtÞjq1;3 dt a a Z c f1þ ðtÞ dt; 6 jxðcÞjq1;1 jyðdÞjq1;2 jzðeÞjq1;3 a

where jyðdÞj ¼ maxa6t6b jyðtÞj and jzðeÞj ¼ maxa6t6b jzðtÞj. Substituting the above inequality into (14), we obtain

1 6 jxðcÞjq1;1 p1 jyðdÞjq1;2 jzðeÞjq1;3

Z

c

1

ðr 1 ðtÞÞ1p1 dt a

p1 1 Z c  f1þ ðtÞ dt: a

From the above inequality we obtain

Z

c

a

f1þ ðtÞ dt P jxðcÞjp1 q1;1 jyðdÞjq1;2 jzðeÞjq1;3

Z

c

1

ðr1 ðtÞÞ1p1 dt

1p1 :

ð15Þ

a

Similarly, by using xðbÞ ¼ 0, we can show that

Z c

b

f1þ ðtÞ dt

p1 q1;1

P jxðcÞj

jyðdÞj

q1;2

q1;3

jzðeÞj

Z c

b

1 1p1

ðr 1 ðtÞÞ

!1p1 dt

:

ð16Þ

X. Yang et al. / Applied Mathematics and Computation 219 (2012) 1670–1673

1673

Since the function hðxÞ ¼ x1p with p > 1 is convex for x > 0, the Jensen’s inequality

 x þ y 1 < ½hðxÞ þ hðyÞ h 2 2

implies that

Z

c

1 1p1

ðr1 ðtÞÞ

1p1 dt

þ

a

Z

b

1 1p1

ðr 1 ðtÞÞ

!1p1 dt

>2

p1

c

Z

b

1 1p1

ðr1 ðtÞÞ

b

f1þ ðtÞ dt

a

> jxðcÞj

p1 q1;1

dt

:

ð17Þ

a

Now, (15)–(17) imply that

Z

!1p1

q1;2

jyðdÞj

jzðeÞj

q1;3

2

p1

Z

b

ðr 1 ðtÞÞ

!1p1

1 1p1

dt

ð18Þ

:

a

In an analogous way, for the second and third equation of (7) we can obtain

Z

b

a

f2þ ðtÞ dt > jxðcÞjq2;1 jyðdÞjp2 q2;2 jzðeÞjq2;3 2p2

Z

b

!1p2

1

ðr 2 ðtÞÞ1p2 dt

;

ð19Þ

:

ð20Þ

a

and

Z

b

a

f3þ ðtÞ dt

> jxðcÞj

q3;1

jyðdÞj

q3;2

p3 q3;3

jzðeÞj

2

p3

Z

b

ðr 3 ðtÞÞ

!1p3

1 1p3

dt

a

Raising both sides of inequalities (18)–(20) to the powers e1 ; e2 and e3 respectively and multiplying the resulting inequalities, we obtain

Z 3 Y k¼1

a

!e k

b

fkþ ðtÞ dt

h1

h2

h3

> ½jxðcÞj jyðdÞj jzðeÞj 2

Q

"Z 3 Y k¼1

b

1 1pk

ðr k ðtÞÞ

#ð1pk Þek dt

;

ð21Þ

a

P where hk ¼ ðpk  qk;k Þek  j–k qj;k ej for k ¼ 1; 2; 3. By assumption, Eq. (8) has positive solutions ðe1 ; e2 ; e3 Þ such that hk ¼ 0 for k ¼ 1; 2; 3. Choosing one of the solutions ðe1 ; e2 ; e3 Þ, we obtain from (21) the inequality (9). h Proof of Corollary 1. From the proof of Theorem 1, we see that (10) implies e1 ¼ e2 ¼ e3 ¼ 1 is a positive solution of (8). Now Corollary 1 is a direct consequence of Theorem 1. h Proof of Corollary 2. If (I) holds, then the three equations of (8) reduce to equivalent equation. In this case, it is easy to see that (8) has a positive solution ðe1 ; e2 ; e3 Þ. If (II) holds, let a1 ; a2 and a3 be the first, second, and the third row vector of the matrix A3 respectively. Then it is not difficult to verify that

a1 ¼ a1  a2 ; a2 ¼ a2  a3 ; a3 ¼ a3  a1 ; where  is the cross product operator. Since rank A3 ¼ 2, at least one of the vectors a1 ; a2 and a3 is nonzero. If a1 is positive, we see that a1  a2 ¼ a1 – 0, which implies that the vectors a1 and a2 are linearly independent. Since rank A3 ¼ 2, we see that the vector a3 is a linear combination of a1 and a2 . From a property of cross products, we see that a1 ? a1 and a1 ? a2 , hence a1 ? a3 . This implies that ðe1 ; e2 ; e3 Þ ¼ a1 is a positive solution of (8). The other cases can be discussed similarly. Combining the above discussion, we see that under the assumption (II), (8) has a positive solution ðe1 ; e2 ; e3 Þ such that the inequality (9) holds. h Acknowledgements The authors thank an anonymous referee for his kind corrections and suggestions to the original manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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