Oscillation Theorems for a Class of Nonlinear Second Order Differential Equations with Damping

(1)

Oscillation Theorems for a Class of Nonlinear Second

Order Differential Equations with Damping

Xiaojing Wang, Guohua Song

School of Science, Beijing University of Civil Engineering and Architecture, Beijing, China Email: [email protected]

Received November 8, 2012; revised December 7, 2012; accepted December 14,2012

ABSTRACT

The oscillatory behavior of solutions of a class of second order nonlinear differential equations with damping is studied and some new sufficient conditions are obtained by using the refined integral averaging technique. Some well known results in the literature are extended. Moreover, two examples are given to illustrate the theoretical analysis.

Keywords: Nonlinear Differential Equations; Damping Equations; Second Order; Oscillation Solutions

1. Introduction

In this paper, we are concerned with the oscillatory be-havior of solutions of the second-order nonlinear differ-ential equations with damping

 



 





 





r t  x t k x t





 



 



 





p t k x t

x t

  

  



C

 , ,k f g, 



,



C R R

1 2 1

, , ,

c c c

 



 



q t f x t g

     

, ,

r t p t q t

0

0,t t 0,

 





t0, ,R



(1.1)

where and

.

In what follows with respect to Equation (1.1), we shall assume that there are positive constants  and 2 satisfying

(A1) r t

 

0 and for all ; 0 

 

x 0 x0

1

c

xf



t



(A2) c  x

 

 for all x; (A3)  1 0 and k

 

y

 

for all

2

1yk y



 yR;

(A4) q t

 

0 and 0c2g x t





 



;

 

(A5) 2

f x

x x0

   

 



 



0

r t x t q t f x t 

 

 

   

(

     



 



0

r t x t p t x t q t f x t 

 

 

 



 





 







 



 



 



 



0,

r t x t k x t p t k x t

q t f x t



 

 

 

0 

  for all .

We shall consider only nontrivial solutions of Equa-tion (1.1) which are defined for all large t. A solution of Equation (1.1) is said to be oscillatory if it has arbitrarily large zeros, otherwise it is said to be nonoscillatory. Equation (1.1) is called oscillatory if all its solutions are oscillatory.

The oscillation problem for various particular cases of Equation (1.1) such as the nonlinear differential equation

 _{, (1.2)}

the nonlinear damped differential equation

 _(1.3)

and

(1.4)

have been studied extensively in recent years, see e.g. [1-21] and the references quoted therein. Moreover, in 2011, Wang [22] established some oscillation criteria for Equation (1.1) firstly, some new sharper results are ob-tained in the present paper.

An important method in the study of oscillatory be-haviour for Equations (1.1)-(1.4) is the averaging tech-nique which comes from the classical results of Wintner [19] and Hartman [18]. Using the generalized Riccati technique and the refined integral averaging technique introduced by Rogovchenko and Tuncay [20,21], several new oscillation criteria for Equation (1.1) are established in Section 2, we also show some examples to explain the application of our oscillation theorems in Section 2. Our results strengthen and improve the recent results of [1] and [21,22].

2. The Main Results

Following Philos [10], let us introduce now the class of functions  which will be extensively used in the se-quel. Let









0 , : 0

D  t s t s t and D



 

t s, :t s t0



.



;



H

The function C D R is said to belong to the class  if

 

, 0

H t t  for ; on ;

1) 0 0

2)

(2)

3) There exists a function h t

 

,s C D R



,



such that



 

,

_{ }

H t s

h t,s H t s,

s



 

 .

In this section, several oscillation criteria for Equation (1.1) are established under the assumptions (A1)-(A5). The first result is the following theorem.

Theorem 2.1. Let assumption (A1)-(A5) be fulfilled and H . If there exists functions R,C t





₀,



,R



such that

 

rR C t1 0, , R



 

and

0 ₀

,

0 inf lim inf , ,

s t t

H t s H t t

 

 

 

   

 

 

 

   

(2.1)

0

2

lim sup t d ,

t t

s s s r s  







 

0, 1

T t

(2.2)

and for any   

 

_{ }

,

   

1 1

     

2

1

lim sup , , d .

, 4

t

T t

c

T H t s Q s s r s h t s s

H t T

 

 



 

 _  _

 



(2.3)

where

 

Q t  t  (2.4)

 

   

_{ }

 

_{ }

2

2 1

1 1 1 1

4

2

exp t d ,

p t

t R t r t p t R t r t R t

c c c c r t

R s p s

t s

 



   

 __ ___ _ _ _ 

     

   

 

   

 _



_  _ _

 

2 2

c q

1 1 2

c _  r s _

 

 

s max



 

s , 0



  

(2.5)

and  , then Equation (1.1) is

oscil-latory.

Proof. Let x t

 

x t 

be a nonoscillatory solution of Equation (1.1). Then there exists a T0 0 such that

for all . Without loss of generality, we may assume that on interval



. A simi-lar argument holds also for the case when

t



0 tT₀

 

0

x t  T₀,



 

x t is even-tually negative. As in [1], define a generalized Riccati transformation by

 

r t

 



x t

 

_{ }



k x t



 



   

v t t r t R t

x t

    

   

 

 

 

(2.6)

for all tT0, then differentiating Equation (2.6) and

using Equation (1.1), we obtain

  



 



 



 

   



 





 



 

   



 





 



 

     

2

t p t k x t t

v t v t

t x t

t g x t f x t x t

t x t k x t x t

x t

t R t r t  



 

  



  

  _ _

 

t q

t r 

 



(2.7)

In view of (A1)-(A5), we get

 



  

_{ }



 





 

 

_{ }

   

 

_{ }

 



     

 

_{ }

 

   

 

_{ }

   

_

_{ }

_

_

 

_{ }

_

 



 



_{ }



 



 

_{ }

     

   

_{   }

 

   

_{ }

 



 



_{ }



 



2

2 2 2

1

2 2

1 1

1

2 1

1 1

2 4

4

x t t r t x t k x t

t

v t c t q t t R t r t

t x t

x t k x t

t t p t t r t p t

t r t t v t

t r t x t x t x t r t

x t k x t

t t p t t r t

v t

t cr t c x t

 



  



    

  

 

   



 

  

 _

     _ _

2 2

t p t k

t x

c t q

2 2

t R t r t c t q t

 

v t

t



R 

   

 

 _ _       

  _ _

 



   

 

  _ _ 

 

     

   

_{   }

 

   

_{ }

 



 



_{ }



 



 

_{ }

     

   

_{   }

 

   

_{ }

 

_{   }

 

_{ }

 

2 2

1 1 1

2

t R t r t c t q

t R t p

Q t

t c c

 



 

  _ _ 



    

 

2 1

2 2

1 1

2 2

1 1

2 2

1 1

2

4 2

p t r t

x t k x t

t t p t t r t p t

t R t r t c t q t v t

t cr t c x t r t

t t p t t r t v t p t

t v t R t

t cr t c t r t r t

t r



    

  

 

    



  

 

 

 

   

 

 _ _       

 

 

 

  

    _  _ 

   

 

 

1 1

     

2

 

1 1

     

2

1 1

v t v t Q t v t

t c  t r t c  t r t

 

   

 

(3)

for all with



defined as above. Then we obtain

0

T Q t



   

t

 

1 2

 

v t c_{1 1} t r t

Q t v t

 



   . (2.8)

On multiplying Equation (2.8) (with t replaced by s) by H t s

 

,

0

t T T

   

 

_{   }

, integrating with respect to s from T to t for , using integration by parts and property 3), we get

 

   

 

_{     }

   

 

_{   }

 

2

1 1

2

1 1 2

1

, d , d , d

1

, d , , d

, d .

t t t

T T T

t t

t

T T T

H t s Q s s H t s v s s H t s v s s

c s r s

1 1

, t , ,

T

H t s v s v s H t s H t s v s s

c s r s

v s

H t T v T v s h t s H t s H t s

 c s r s s  

 



  

   

 

    

 

 



1



Then, for any  

   

 

_{   }

 

     



  

_{     }

2

1 1 1 1

2 2

1 1

, 1

, d , , d

2

1 ,

, d d ,

4

t t

T T

t t

T T

H t s

H t s Q s s H t T v T v s c s r s h t s s

c s r s

H t s c

s r s h t s s v s s

c s r s

   



 _

 

 

    

 

 



 



and, for all t T T0,

   

     

   

 

_{     }

     



  

_{     }

2

1 1 1 1

, 1 ,

d d

2

H t s

2 1 1

2

1 1

, , d

4

1

, ,

t

T

t t

T T

c

H t s Q s s r s h t s s

H t s

H t T v T v s c s r s h t s s

 _

 

  

 _ 

 

 

 

    



v s s

c s r s c s r s

  

 

 

 

(2.9)

Furthermore,

 

   

1 1

     

2

 



1 1

  

     

2 d .

1 ,

1 1

, , d

, 4 ,

t t

T T

H t s c

H t s Q s s r s h t s s v T

H t T H t T c s r s



 _

 



 _  _ _

 

 



v s s

Now, it follows that

   

     

 

_{ }



  

_{     }

1

2

, 4

1 , 1

lim inf d .

,

T t

t

T t

2 1 1

1 1

1

lim sup



tH t s Q s,   c s r s h t s, ds

 

H t T

H t s

v T v s s

H t T c s r s

  



 

 



 

_

(2.10)

From (2.3) and (2.10), we have

 



1 1

  

     

2

1 , 1

lim inf d

,

t T t

H t s

v s s

H t T c s r s

  





 

_

v T  T

0

TT 1

for all and  

 

v T  T T₀

. Obviously,

for all T (2.11)

and





     

 



    





0

2

0

1 1

0 0

, 1

lim inf d

,

1

t

T t

H t s

v s s

H t T s r s

c

v T T



 _





   





(2.12)

Now, we can claim that

 

   

0

2

d

T

v s

s s r s 



 



, (2.13) Otherwise,

 

   

0

2

d

T

v s

s s r s 



 



. (2.14)

By (2.1), there exists a positive constant  such that

 

0 ₀

,

inf lim inf 0 ,

s t t

H t s

H t t 

 

 

 _{  }

 

 

 

2 1

T T

,

(4)

 

   

 

, 1

 

H t T H t,t₀  for all tT2.

0

On the other hand, by (2.14) for any   1 0

T T

 

   

, there ex-ists a such that

0

2

d

t

T

v s

s s r s



 



for all tT1.

Using integration by parts, we obtain





     

 





 

   

 





 





0

0 0

1

2

0

2

0

1

0

, 1

d ,

, 1

,

, 1

d ,

, . ,

t T

t s

T T

t T

H t s

v s s

H t T s r s

H t s v

d ds

H t T s

H t s s

H t T s

H t T H t T



  

  

r 

 

 



 

  _  

 

   







 

 _ _



 





This implies that

 

0

2

0

, 1

d ,

t T

H t s

v s s

H t T



 s r s  for all tT2.  is an arbitrary positive constant, we get Since

 

   





0

 

2

0

, 1

lim inf d

,

t T t

H t s

v s s

H t T  s r s





 ,

which contradicts (2.12), so (2.13) holds, and from (2.11)

 

   

 

0 0

2 2

d d

T T

s v s

s s

s r s s r s



 

 _ 

  



  

 

1



, n , ,

,

which contradicts (2.2), then Equation (1.1) is oscilla-tory.

H

Now, we define t s  t s  t sD

2

n H

, here . Evidently,  

  





and





3 2

, 1 n , ,

h t s  n ts  t sD .

Thus, by Theorem 2.1, we obtain the following result. Corollary 2.1. Let assumption (A1)-(A5) be fulfilled. Suppose that (2.2) holds. If there exist functions





0



, , ,

RC t  R such that

 

rR C1





t0,



,R



,





1

 

1 1



1

     

2 3

 

1

sup d

4

t n n

n T

c n

t s Q s s r s t s s T

t

 

 

 

 

 _ 

   

 

 



1 lim

t

,

where and are defined as in Theorem 2.1, then Equation (1.1) is oscillatory.

 

Q t 

 

t Example 2.1. Consider the nonlinear damped

differ-ential equation

 





 



 





 



2 3 4 2 2 2 2 2

2 2 2 2 2 6 6 sin 1 1 0

1 1

x

x t

t t t t t t x t x t x t

x t

 

     



        

 

 



,



x   t1

1 e

2 2

x t x t

 _ _ 

 _ _ 

  .



where and , 1

2

c , c₁c₂1,

 

2

_{ }

1 1

f x

x t 2 1

x  

2

   .

The assumptions (A1)-(A5) hold. If we take   , and , then

3

n R t

 

t 

 

t 1, and



2 2 2

2 3 6 sin

Q t   t  t t

 

.

A direct computation yields that the conditions of Co- rallary 2.1 are satisfied, Equation (1.1) is oscillatory.

As a direct consequence of Theorem 2.1, we get the following result.

Corollary 2.2. In Theorem 2.1, if condition (2.3) is replaced by

   

1 1

     

2

 

1 1

, , d ,

, 4

t

T

c H

lim inf

t _T t s Q s    s r s h t s s  T

 _  _

 

 



H t 

where and are the same as in Theo-rem 2.1, then Equation (1.1) is oscillatory.

   

,

Q t  t 

 

t

1

Theorem 2.2. Let assumption (A1)-(A5) be fulfilled. For some  , if there exist functions







0



, , ,

R C t  R such that

 

1







0, ,

C t R

 

 

rR

and

   

 

   

_{ }

     

0

2

1 1 1 2

0 1

1

lim sup , , , d ,

, 2 2

t t t

p s s c

H t s Q s H t s s r s h t s s

H t t c r s

  _{  }



 

   

 

 



(5)

where Q t

 

is the same as in Theorem 2.1, H 

and

 

1 1

d t

R s s



 

 







 

2 exp

t

c

   (2.16)

Then Equation (1.1) is oscillatory.

t

tion (1.1

be a nonoscillatory solution of Equa-

). Then there exists a T0t0 such that

 

0

x t  for all 0 ithout loss of rality, we may assume that

tT . W gene

 

0

x t  on interval



T . A simi-lar argument hol



0,

ds also for the case when x t

 

is even-tu

on Proof. Let

ally negative.

Define the functi v t

 

as in (2.6). Using (A1)-(A5) and (2.7), we have

x

 

_{   }

 

   

_{ }



 



   



 

_{ }

 

 



     

 

_{ }

 

_{   }

_{     }

 

_{   }

_{     }

2

2 2 2

1

2

1 1 1 1 1

2

1

t p t k x t t r t x t k x t

t

v t c t q t t R t r t

t x t x t

t R t p t

Q t v t v t

t c c r t c t r t

p t

Q t v t v t

c r t c t r t

 



  

 



   

 

  



1 1 1

2

v t      _ _

  

  _   _ 

 

   

(2.17)

here Q t

 

is the same as in Theorem 2.1. On the nd, s

w

other ha ince the inequality

2

m n

2 2

ml nl l

n

  

holds for all n0 and ,m lR. Let



 



   

 

1 1 1

1

, ,

t

m n l v t

c r t c r t t



 

    ,

we get from (2.17) that

 

₁p2

   

t

_{ }

 t

 

2

 

_{   }

1 1 1

0

,

2 2

.

v t

Q t v t

c r t c t r t

t T

 



   



(2.18)

On multiplying (2.18) (with t replaced by s) by

 

,

H t s , integrating with respect to s from T to t for 0

T and 1

t T   , using integration by parts and ), we g

property 3 et

   

_{ }

   

 

_{     }

     



  

_{     }

2 1

1

2

1 1 1 1

2 2

1 1

, d

2

, 1

, 2 , d

2 2

1 ,

, d d ,

2 2

t

T

t t

T T

p s s

H t s Q s s

c r s

H t s

1 1 t

T

H t T v T v s c r s s h t s s

c r s s

H t s c

r s s h t s s v s s

c r s s

 

 



 _

 

 



 

 

 

 

  

 

 



 



This implies that

 



   

 

     

 



  

_{   }

 

     

2

1 1 1 2

1

2

1 1

2

1 1 1 1

, , , d

2 2

1 ,

, d

2

, 1

2 , d

2 2

t

T

t

T

p s s c

t

T H t s

H t T



Q s H t s r s s h t s s

c r s H t s

v T v s s

c r s s

H t s

v s c r s s h t s s

c r s s

   _



 

 

 

 

 



 

 

 

 

 

 



Using the properties of H t s

 

, , we have

   

 

   

_{ }

   



  



  

 

   

0

2

1 1 1

1

0 0 0 0 0 0

, ,

2 2

, , , .

t T

p s s c 2

, d

H t s Q s H t s r s s h

c r s

H t T v T H t T v T H t t v T

   _

 

  



fore,

t s s

 

 

 

(6)

   

 

   

_{ }

   

 

   

_{ }

 

   

 

   

_{ }

 

   

 

0

0 0

0

0 0

2

1 1 1 2

1

2

1 1 1

1

2

1 1 1

1

0 0

, ,

2 2

, ,

2 2

, ,

2 2

, d

t

T

t

T

t

p s s c

H t s Q s H t s r s s h

c r s

p s s c 2

2

, d

t s s

H t s Q s H t s r s s

c r s

p s s c

h t s s

H t s Q s H t s r s

c r s

H t t Q s s v T

   _

 

  

  

 

 _  _

 



s h t s s

 

 

 

 

 

 

 

 

 

for all tT0, and so

 

   

 

   

 

   

 

0

0 0

2

1 1 1

0 1

lim sup , ,

, 2

d ,

t

t t

T

t

p s s c 2

, d

2

H t s Q s H t s r s

H t t c r s

Q s s v T

   _



 

   



s h t s s

 

 

 

which contradicts with the assumption (2.15). This

com-

pletes the proof of Theorem 2.2.

Let H t s

  

,  t s

 

n1, ,t sD , from Theorem 2.2, lt.

Corollary 2.3. Let assumption (A1)-(A5) be fulfilled. If

we obtain the next resu

there exist functions R, C t





0,



,R



such that

 

1







0, ,

C t R ,

and

rR  









1

 

1 2

   

_{ }

1 1 2

   



3

1

d

2 2

n p s s c n n

t s Q s s r s t s s

c r s

   



 

    

     

 _ _ 

 _ _ 

 



olds for some integer n2 and 1 1

1 lim sup _n t

T t t 

 

x



,



h   , where Q t   , 1

and 

 

t are defined as heorem then Equ (1.1) illatory.

Example 2.2. Consi

in T 2.2, ation

is osc

der the nonlinear damped differ-ential equation

 





_{ }

 

_{ }

 

_{ }

 



 



2 2

2 2 2

2 2

2

1 1

1 1 1

3 1

2 1 1 0.

8 2

2

2 x t x t x t

t t t

x t x t x t

t x t x t

x t



 

  

 

 _ _ _  _ _

 

 

  _

   _  _ _  

 

(2.19)

Evidently, for all   and t1,

 

2 1 x t 2 c1

 

 

we

c c 

have

  ,

  

and

 

2 1

2

1

1 1

2

f x

x   x t    .

Let R t

 

0,n3, then

and

 

t 1

  1 2

2 4

Q t   t .





 

   

_{ }

   











2 2

1 1 1 1 3

1

2 2

( 1)

d

2 2

4 1 d ,

n p s s c n n

Q s s r s t s s

c r s

t s s s

    _



 

1

2 1 1 lim sup

1

im sup 2

t

n _T

t

t s t

t  



 _  _ _  _ 

 _ _ 

   

 

 _ _  _{ }

 

erefo n (2 s os ry by Corallary

2.

ption (A1)-(A5) be fulfilled

and H

 









Th re, Equatio .19) i cillato . If there exist functions R, C t0, ,R

ch that







3.

Theorem 2.3. Let assum

su (2.1) holds and

 

1

0, ,

C t  R , and

fo for some 1

rR 

r all tt0, any T t0, and   ,

 

   

 

   

 

1 2 1 1

     

2

1

, , , d ,

, 2 2

t T

p s s c

T H t s Q s H t s s r s h t s s

H t T c r s

   

      

 

 



(2.20)

 

Q t and 

 

t are the same as in Theorem 2.2

 



n (1.1

Proof. The proof of this theorem is similar to that of .

lim sup

t

where

and  max



 , 0



. If (2.2) is satisfied, then

Equa .

Theorem 2.1 and hence is omitted

 

s s

) is oscillatory

(7)

fulfilled except the condition (2.20) be replaced by

 

_{ }

   

 

1 2

   

_{ }

1 1

 

1

lim inf , ,

, 2

t T t

p s s c

T H t s Q s H t s

H t T c r s

   

 





_



   

2

, d , 2 s r s h t s s

 



 

 

en Equation (1.1) is oscillatory. th

Remark 2.1. If we take f x

 

x, then the condition

q

Rema

 

t 0 is not necessary.

rk 2.2. If we take g x t



 



1,

 

to Theore

k x x

    

Th m 9 and 10 o

, then

eorem 2.3 and 2.4 reduce f [21]

with  1 1, respectively.

Remark 2.3. If replace (A5) and (2.6) by f

 

x ex-ists, f

 

x 2 0 for x0 and define

 

   



 

_



_{ }



_

 



 

v t t r t R t

f x t

 

   

 

 

respectively, we can obtain similar oscillation results th

3. Acknowledgements

the National Natural

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This work was supported by

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