Lyapunov-Type Inequalities for the Quasilinearq-Difference systems

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 2 Ver. IV (Mar. - Apr. 2017), PP 54-59 www.iosrjournals.org

Lyapunov-Type Inequalities for the Quasilinearq-Difference systems Jiannan Song, YunfeiGao, ChengminHou* (Department of Mathematics, Yanbian University, Yanji 133002, P. R. China)

Abstract: Using the Hӧlder inequality, we establish several Lyapunov-type inequalities for quasilinear q-difference equation and q-difference systems.

Keywords: Lyapunov-type inequalities, q-difference equation, q-difference systems, Hӧlderinequality I.

Introduction

The Lyapunov inequality and many of its generalizations have proved to be useful tools in oscillation theory, disconjugacy, eigenvalue problems, and numerous other applications for the theories of differential and difference equations. The well-known inequality of Lyapunov[1]states that a necessary condition for the boundary value problem y ''  q(t ) y  0, y(a)  y(b)  0, to have nontrivial solutions is that b | g (t ) | dt  4 .  a

ba

There are several different proofs of this inequality since the original one by Lyapunov[1]. In the last few years independent works appeared generalizing Lyapunov’s inequality for the p-Laplacian, by usingHӧlder, Jensen or Cauchy-Schwarz inequalities. For the case of a single equation, see for example [2-9]. For systems there are a few results.

In the paper, we consider boundary problem of the following quasilinear q-difference equation [10,11].

 Dq (r (t ) | Dq (u (t )) | p 2 ( Dq (u (t )))  f (t ) | u (t ) | p  2 u (t ), t  (0,1), 0  q  1.  (1)   u (0)  u (1)  0, u (t )  0, t  (0,1). and boundary problem of quasilinearq-difference system

 Dq (r1 (t ) | Dq (u (t )) | p1 2 ( Dq (u (t )))  f1 (t ) | u(t ) |1 2 u(t ) | v(t ) |2 , t  (0,1), 0  q  1,    p 2  2   Dq (r2 (t ) | Dq (v(t )) | 2 ( Dq (v(t )))  f 2 (t ) | v(t ) | 2 v(t ) | u(t ) | 1 , t  (0,1), 0  q  1, (2)  u (0)  u (1)  v(0)  v(1)  0, u (t )  0, v(t )  0, t  (0,1).   For the sake of convenience, we give the following hypothesis (H 1) and (H2) for (1) and hypothesis (H3) for (2): (H1) r (t ) and f (t ) are real-valued functions and r (t )  0 for all t  R . 1  p, p1  ， satisfy

(H2)

1 1   1. p1 p

1  p1 , p2  , 1 ,  2 , 1 ,  2  0, satisfy

1  2 p1



p2

 1 and

1 p1



2 p2

 1.

We recall some concepts for q-difference operator. Throughout this paper, we assume The q-derivatives Dq f and

q  (0,1).

 q

D f of a function f are given by

f ( x)  f (qx) f (q 1 x)  f ( x)  ( Dq f )( x)  , ( Dq f )( x)  , if x  0.

(1  q) x (1  q) x ( Dq f )(0)  f '(0) and ( Dq f )(0)  q1 f '(0) provided f '(0) exists.

The q-integrals is defined by



a

0

DOI: 10.9790/5728-1302045459



f ( x)d q x  (1  q)a  f (aq n )q n , n 0

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Lyapunov-Type Inequalities for the Quasilinear q-Difference systems



b

a

b

a

0

0

f ( x)dq x   f ( x)dq x   f ( x)dq x.

The q-analogue of Leibnitz rule is given by

Dq f ( x) g ( x)  g ( x) Dq f ( x)  f (qx) Dq g ( x).

An immediate consequence is the q-integration by parts rule (see[12]).



b

a

b

g ( x) Dq f ( x)dq x  f (b) g (q 1b)  f (a) g (q 1a)   f ( x) Dq g ( x)d q x. (*) a

II.

Lyapunov-Type Inequalities For Q-Difference Equation (1)

In the section, we establish Lyapunov-type inequality for q-difference equation (1).

Denote t

 (t )  (  r

 p1 p

0

1

 (t )  (  r

1 p1

t

( s)d q s)

p p1

( s)d q s)

t

 ( r

1 p1

0

1 p1 1

( s) d q s)

1 p1 1

,

.

Theorem2.1. Suppose that hypothesis (H1) holds. If boundary value problem (1) has a solution. Then one has the following inequality: 1

 (t ) (t )

  (t )   (t ) f



0

where

f  (t )= max  f (t ),0.

(t )d qt  1, (3)

Proof.By (1) and the q-integration by parts rule (*) 1

  Dq (| r (t ) || Dqu (t ) | p  2 Dqu (t ))u (t )d qt 0

1

 u (t )[r (q 1t ) | Dqu (q 1t ) | p  2 Dqu (q 1t )] 10   r (t ) | Dqu (t ) | p 2 Dqu (t ) Dqu (t )d qt 0

1

1

  r (t ) | Dqu(t ) | dqt   f (t ) | u(t ) | dqt. (4) p

0

p

0

By the boundary condition of (1), we have t

t

1 p

t

1 p

| u (t ) | |  Dq u ( s )d q s |  (  | Dqu ( s ) | d q s )  (  r (s )r (s ) | Dqu (s ) | d q s ) p p

p

0

t

 ( r

 p1 p

0

p

0

( s)d q s )

p p1

t



t

0

0

1

t

t

r ( s ) | Dqu ( s ) | p d q s =(  r 1-p1 ( s )d q s ) p1 -1  r ( s ) | Dqu ( s ) | p d q s 0

0

= (t ) r (s) | Dqu(s) | dq s, p

0

and 1

1

1

| u (t ) | |   Dqu ( s)d q s |  (  | Dqu ( s) | d q s)  (  r p

p

t

p

t

t

1-p1

( s)d q s)

1 p1 -1

1

 r ( s) | D u (s) | d s t

p

q

q

1

= (t )  r ( s) | Dqu ( s) | p d q s. t

Thus

| u (t ) | p 

 (t ) (t ) 1 r ( s) | Dqu ( s) | p d q s.  0  (t )   (t )

By (4), we have



1

0

p

f  (t ) | u (t ) | d qt 1 1  (t ) (t ) 1  (t ) (t )  f (t )d q t  r (t ) | Dq u (t ) | p d qt   f  (t )d qt  f (t ) | u (t ) | p d qt 0  (t )   (t ) 0 0  (t )   (t ) 0



1

DOI: 10.9790/5728-1302045459

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Lyapunov-Type Inequalities for the Quasilinear q-Difference systems 1  (t ) (t )  f (t )d qt  f  (t ) | u (t ) | p d qt. (5) 0  (t )   (t ) 0



1

Next, we prove

that



If (6) is not true, then

p

1

f  (t ) | u (t ) | d qt  0. (6)

0



1

0

p

f  (t ) | u (t ) | d qt  0. (7)

From (4) and (7), we have p

1

p

1

p

1

0   r (t ) | Dqu(t ) | d qt   f (t ) | u (t ) | d qt   f  (t ) | u (t ) | d qt  0.

0

0

 

0

It follows Dq u q t  0, n  0,1, 2,, we obtain that n

u  t   0,

t   0,1 which contradicts the

for

condition of (2). Therefore, from (5), we may see that (3) holds. The proof is completed.

Note that (

  2

) 2   , one has following corollary 2.1.

Corollary 2.1.Suppose that hypothesis (H1) is satisfied. If (1) has a solution inequality:



1

0

III.

u(t ). Then one has the following

1

( (t ) (t )) 2 f  (t )d qt  2.

Lyapunov-Type Inequalities For Q-Difference Equation (2)

Denote t

i (t )  ( r

1 pi 0 i 1

i (t )  ( r t

1 pi i

( s)d q s ) ( s)d q s)

1 pi 1 1 pi 1

, i  1, 2. (8) , i  1, 2. (9)

Theorem3.1. Suppose that hypothesis (H2) is satisfied. If system (2) has a solution the following inequality: 11

 (t )1 (t )  ( 1 f (t )dqt ) 0  (t )   (t ) 1 1 1 1

  (t )2 (t )  ( 2 f1 (t )dqt ) 0  (t )   (t ) 2 2  fi (t )  max{ fi (t ), 0}, for i  1, 2.

1 2

1

where

p12

p1 p2

(u(t ), v(t )). Thenone has

 1 (t )1 (t )  ( f (t )dqt ) 0  (t )   (t ) 2 1 1

1 2

1

 2 2

 (t )2 (t )  ( 2 f (t )d qt ) 0  (t )   (t ) 2 2 2 1

p1 p2

p22

 1.

Proof. Similar to(4), we have 1

1

  r (t ) | D v(t ) | d t   f (t ) | v(t ) | |u(t ) | d t, (11) |  D u(t )d t |   (t ) r (t ) | D u(t ) | d t ,   (t ) r (t ) | D u(t ) | d t ,  (t ) (t ) | u (t ) |  r (t ) | D u (t ) | d t. (12)  (t )   (t )  0 1

r1 (t ) | Dqu(t ) | p1 dqt   f1 (t ) | u(t ) |1 |v(t ) |2 dqt, (10) p2

0 2

| u(t ) | p1 | u(t ) | p1

t

0

q

q

1

t 1

q

1

q

q

p1

q

q

1

1

0

1

p1

0 1

p1



2

2

0

t

1

1

q

p1

q

0 1

1

1

1

p1

0 1

q

1

1

0

0

q

f1 (t ) | u (t ) | p1 d qt  11  r1 (t ) | Dqu (t ) | p1 d qt  11  f1 (t ) | u (t ) |1 | v(t ) | 2 d qt 1

 11  f1 (t ) | u (t ) |1 | v(t ) | 2 d qt 0

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Lyapunov-Type Inequalities for the Quasilinear q-Difference systems 1

1

1

2

 11 ( f | u(t ) | dqt ) (  f (t ) | v(t ) | d qt ) p2 . (13) 0



1

0

 1

p1

p1

0

 1

p2

1

1

f (t ) | u(t ) |  12  f1 (t ) | u(t ) |1 | v(t ) |2 dqt  12  f1 (t ) | u(t ) |1 | v(t ) |2 dqt  2

p1

0

0

1

1

2

1

 12 ( f1 (t ) | u(t ) | p1 dqt ) p1 (  f1 (t ) | v(t ) | p2 d qt ) p2 . (14) 0

where 11

0

1  (t ) (t ) 1 (t )1 (t )  1 f1 (t )d qt , 12   1 f  (t )d qt.

0  (t )   (t ) 0  (t )   (t ) 2 1 1 1 1



1

Similar to the proof of(12),from(10)(11), we have

v(t )



p2

2 (t )2 (t ) 1 r (t ) | Dq v(t ) | p d qt. 2 (t )  2 (t ) 0 2 2

It follows from above form and the Hӧlder inequality that



1

0

1  2 (t )2 (t )  f1 (t )d qt  r2 (t ) | Dq v(t ) | p d qt 0  (t )   (t ) 0 2 2

f1 (t ) | v(t ) | p2 d qt  

1

2

1

 M 21  f 2 (t ) | u (t ) |1 | v(t ) |2 d qt 0

1

1

2

1

 M 21 (  f (t ) | u (t ) | d qt ) (  f (t ) | v(t ) | d qt ) p2 .

0



1

0

 2

p1

p1

 2

0

p2

1

1

(15) 2

1

f (t ) | v(t ) |  M 22 (  f (t ) | u(t ) | d qt ) (  f (t ) | v(t ) | d qt ) p2 . (16)  2

p2

 2

0

p1

p1

 2

0

p2

1  (t ) (t )  2 (t )2 (t )  2 f1 (t )d qt , M 22   2 f  (t )d qt. (17) where M 21   0  (t )   (t ) 0  (t )   (t ) 2 2 2 2 2 1

Similar to (5), we have

 

1

0 1

0

1

f1 (t ) | u (t ) p1 | d qt  0,  f 2 (t ) | u (t ) p1 | d qt  0, 0

1

f (t ) | v(t ) | d qt  0,  f (t ) | v(t ) | d qt  0.  1

p1

0

 2

(18)

p2

From(14)-(16), (18),we have 11

1 2

1 2

 2 2

 1.

Corollary 3.1. Suppose that hypothesis (H2) are satisfied. If (2) has a solution (u(t ), v(t )). Then one has M11

p12

M12

p1 p2

M 21

p1 p2

M 22

p22

the following inequality: 1

1

(  f1 (1 (t )1 (t )) 2 d qt )

1 2

p1 p2

0

1

1 2

(  f (1 (t )1 (t )) d qt ) 0

 1

1

1

(  f 2 ( 2 (t )2 (t )) 2 d qt )

1 2

p1 p2

0

1 2

p1 p2

1

1 2

(  f ( 2 (t )2 (t )) d qt ) 0

 2

1 2

p1 p2

 2 p2 1  p12  p1 p2 .

Next,we consider the quasilinear q-difference system involving the ( p1 , p2 , …, pm )  Laplacian:

 Dq (r1 (t ) | Dqu1 (t ) | p1 2 ( Dqu1 (t )))  f1 (t ) | u1 (t ) |1 2 | u2 (t ) | 2 … | um (t ) | m u1 (t ),    p 2   2  Dq (r2 (t ) | Dqu2 (t ) | 2 ( Dqu2 (t )))  f 2 (t ) | u1 (t ) | 1 | u2 (t ) | 2 … | um (t ) | m u2 (t ), (19)    D  (r (t ) | D u (t ) | pm  2 ( D u (t )))  f (t ) | u (t ) |1 | u (t ) | 2 … | u (t ) | m  2 u (t ). q m q m m 1 2 m m  q m with boundary value conditions:

ui (0)  ui (1)  0 , ui (t )  0 , t  (0,1) , i  1, 2, , m. (20) We give the following hypothesis (H3).

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Lyapunov-Type Inequalities for the Quasilinear q-Difference systems

ri (t ) and fi (t ) are real-valued functions and ri (t )  0 for i  1, 2,, m.

Furthermore, 1  pi   and  i  0 satisfy  m ( i )  1.

(H3)

pi

i 1

Denote t

i (t )  (  r

1 pi 0 i

( s )d q s )

1

i (t )  (  ri1 p ( s)d q s) i

1 pi 1 1 pi 1

t

.

(21)

.

Theorem3.2.Suppose that hypothesis(H3)is satisfied. If system (19) has a solution (u1 (t ), u2 (t ),...., um (t )) satisfying the boundary conditions(20),then one has the following inequality: i j

 1 1 ( )1 ( )   f j ( )dq   0  1 ( )  1 ( ) i 1 j 1   m

Pi Pj

m

 1. (22)

Proof.By(19)(H3) and (20),we can get



1

0

1

m

0

k 1

ri (t ) | Dqui (t ) | pi   fi (t ) | uk (t ) |k d qt , i  1, 2, , m.

It follows from(21)and the Hӧlder inequality that t

| ui (t ) | pi  i (t ) ri ( ) | Dqui ( ) | pi dq .

0 1

| ui (t ) | pi  i (t ) ri ( ) | Dqui ( ) | pi dq . t

Thus

| ui (t ) | pi 



1

0

where M ij

i (t )i (t ) 1 r ( ) | Dqui ( ) | p d q , i (t )  i (t ) 0 i i

m

k

f (t ) | ui (t ) | dqt  M ij  (  fi (t ) | uk (t ) | ) pk ,  j

pi

k 1

1



pk

0

i (t )i (t )  f (t )d qt , i, j  1, 2,…, m. (23) 0  (t )   (t ) j i i



1

Similar to the proof of the (18),we have



1

0

m

Therefore

 M i 1

i j

m

j 1

pi p j ij

fi  (t ) | uk (t ) | pk dqt  0, i, k  1, 2,…, m.

 1. (24)

It follows from (23) and (24) that (23) holds.

Corollary 3.2. Suppose that hypothesis(H3) aresatisfied. If system (19)has a solution (u1 (t ), u2 (t ),..., um (t )) satisfying(20),then onehas the following inequality: 1  1   2  0 f j ( )((i (t )i (t ))   i 1 j 1   m

m

i j

pi p j

 2.

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