Some Oscillation Properties of Third Order Linear Neutral Delay Difference Equations

(1)

International Journal of Computational Engineering Research (IJCER)

Some Oscillation Properties of Third Order Linear Neutral Delay Difference Equations

A.George Maria Selvam

¹

, M. Paul Loganathan

²

, K.R.Rajkumar

³

1,3

Sacred Heart College, Tirupattur – 635 601, S.India,

2

Dravidian University, Kuppam

I. INTRODUCTION

In this paper, we consider the third order linear neutral delay difference equations from

  a n ^{( )}   

²

x n ^{( )}  p n x ( ) ( ( )))  n   q n x ( ) ( ( ))  n  0

(1) where

n  N n ( )

0

  n n

0

,

0

 1.., ,  n

0is a nonnegative integer, subject to the following conditions

  H a n p n q n

1

( ), ( ), ( )

are positive sequences.

0 0

0 ( ) 1, ( ) , ( ) ,lim ( ) lim ( )

n n

p n p  n n  n n  n  n

 

       

and

0

1

1 ( ) ( )

n

s n

R n a s





   

^as

ⁿ ^{ }

 

0 0

2

1 ( )

( )

^{t n}

n n s n

H q t

a s

  

  

 

   

 

  

^.

 

0

2 2

1 3

( ( )) ( 1)

lim sup ( )(1 ( ( )))

2 4 ( )

n n

s n

KM s a s

H q s p s

sa s

 







  

     

 



^.

We set

z n ( )  x n ( )  p n x ( ) ( ( ))  n

^.

The oscillation theory of difference equations and their applications have received more attention in the last few decades, see [[1]-[4]], and the references cited therein. Especially the study of oscillatory behavior of second order equations of various types occupied a great deal of interest. However the study of third order difference equations have received considerably less attention even though such equations have wide applications. In [[5]- [10]] the authors investigated the oscillatory properties of solutions of third order delay difference equations and in [[11]-[15]]. Motivated by the above observations, in this paper, we investigate the oscillatory behavior of solutions of equation (1).

Let

max lim 

0

( ), ( ) 

x

n n





 



 . By a solution of equation (1) we mean a real sequence x(n) which is defined for all

n  n

₀

 

satisfying (1) for all

n  n

₀. A non-trivial solution x(n) is said to be oscillatory if it is neither eventually positive or eventually negative; otherwise, it is nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.

II. MAIN RESULTS

Lemma 2.1. Let x(n) be a positive solution of equation (1) for all

n  n

₀^such

x n ( )   0, x n ( )  0,

^and

2

x n ( ) 0

 

on

[ , ) n

₁



for some

n

₁

 n

₀. Then for each k with 0 < k < 1, there exists

n

₂

 n

₁ such that

( )

₂

( ) ,

x n n

k n n

x n n

 

 

 

. (2) Proof. From the Lagrange’s Mean value theorem, we have for

n  n

₁, for some k

ABSTRACT

In this paper, we establish some sufficient conditions for the oscillation of solutions of third order linear neutral delay difference equations of the form

 a n ^{( )}

²

^{( ( )} x n p n x ( ) ( ( )))  n  q n x ( ) ( ( ))  n 0

    

^.

(2)

( ) ( ( ))

( ) ( )

x n x n

x k n n



  



; for some k (3)

such that

 ( ) n    k n .

²

x n ( )  0

and

 x n ( )

is non-increasing, which implies that

 x k ( )   x ( ( ))  n

and hence, using equation (3)

( ) ( ( )) ( ( ))( ( )) x n  x  n   x  n n   n

( ) ( ( ))

1 ( ( ))

( ( )) ( ( ))

x n x n

n n

x n x n

 

   

(4)

Apply Lagrange’s Mean value theorem once again for x(n) on

[ , ( )] n

₁

 n

^for

n   n

₁

 ( ) n

^{. Now}

1 1

( ( )) ( )

( ) ( )

x n x n

x c n n



  



for some c such that

n

₁

  c  ( ) n

and

 x c ( )   x ( ( ))  n

which implies

( ( )) ( ( ))( ( )

1

)

x  n   x  n  n  n

. Hence

( ( ))

₁

( ( )) ( )

x n

n n x n

 

 ^ ^



For

K  (0,1),

we can find

n

₂

  n

₁



( ( )) ( ( )) ( ) x n

K n x n

 

 ^



^for

n  n

² (5) From equation (4) and for all

n  n

₂, we have

( ) 1

1 ( ( ))

( ( )) ( )

x n n n

x n K n 

 ^{ }  ^

( )

1 ( ) ( )

n n

K n K n



 

  

 ⁿ ^{( )} 

K  n



Hence,

^{( ( ))}  ^{( )} 

( )

K n

x n

x n n

 _ 

(6)

Lemma 2.2 Let x(n) be a positive solution of equation (1), then the corresponding sequence z(n) satisfies the following condition z(n) > 0,

^ ^{z n}   ^ ^0, ^

²

^{z n}   ^ ^0, ^

³

^{z n}   ^ ⁰

for some

n

₁

 n

₀. Then there exits

n

₂

 n

₁ such that

( )

( ) 2 , z n Mn

z n 



n  n

² for each M, 0 < M < 1.

Proof. We define a function H(n) for

n  n

₂

 n

₁, as

2 2 2

( )

( ) ( ) ( ) ( )

2 M n n

H n  n n z n     z n

(7)

2 2 2 2

( )

( ) ( ) ( ) ( ) ( )

2 M n n

H n z n n n z n  z n

      

(8)

By Taylor’s Theorem,

2 2 2

( )

( ) ( ) ( ) ( ) ( )

2 z n  z n   n n  z n  n n   z n

From (8)

2 2

2 2 2 2

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2

n n M n n

H n z n n n z n  z n n n z n  z n

           

which implies

 H n ( )  0

and

H n (   1) H n ( )  H n ( )

₂

 0

for every

n  n

₂ from (7)

(3)

2 2 2

( )

( ) ( ) ( ) 0

2 M n n

n n z n  z n

   

which implies

( )

( ) 2

z n Mn z n 



^for

n  n

².

Theorem 2.3. Assume that

( H

₁

)

to

( H

₂

)

hold, then equation (1) is oscillatory.

Proof: Suppose, if possible that the equation (1) has a nonoscillatory solution. Without loss of generality suppose that x(n) is a positive solution of equation (1). We shall discuss the following cases for z(n).

(i)

z n ( )  0,  z n ( )  0, 

²

z n ( )  0, 

³

z n ( )  0,

(ii)

z n ( )  0,  z n ( )  0, 

²

z n ( )  0, 

³

z n ( )  0,

Case 1.

z n ( )  0,  z n ( )  0, 

²

z n ( )  0, 

³

z n ( )  0,

Since z(n) > 0and

 z n ( )  0,

then there exists finite limits

lim ( )

n

z n k





. We shall prove that k = 0.

Assume that k > 0. Then for any

 0

, we have

k  z n ( )  k

. Let

(1 )

0 k p

p

 

, we have

( ) ( ) ( ( )) k  x n   k p n x  n

.

( ) ( ) ( )

x n   k p k   m k 

x n ( )  mz n ( )

When

( )

k p k

m k

  

  

. Now from the equation (1) we have

 

 ^{a n} ^{( )}

²

^{x n} ^{( )} ^{p n x} ^{( ) ( ( )} ^ ⁿ  ^{q n x} ^{( ) ( ( ))} ^ ⁿ  ^{a n} ^{( )}

²

^{z n} ^{( )}  ^{q n mz} ^{( )} ^{( ( ))} ^ ^s

        

Summing the above inequality from n to ∞ we get,

 ^{( )}

²

^{( )}  ^{( ) ( ( ))}

t n s n

a t z t m q s z  s

 

 

     

( )

²

( ) ( ) ( ( ))

s n

a n z n m

^

q s z  s



  

Using the fact that

z ( ( ))  n  k

we obtain,

( )

²

( ) ( )

s n

a n z n mk q s





  

which implies

2

1 ( ( )) ( )

( )

s n

z n mk q s

a n





 

   

  

. Summing from n to ∞ we have,

2

1 ( ) ( )

s n s n

( )

t s

z s mk q t

a s

  

  

 

   

 

  

1 ( ) ( )

s n

( )

t s

z n mk q t

a s

 

 

 

   

 

 

Summing the last inequality

n

₁ to



1

( ) 1 ( )

n n s n

( )

t s

z n mk q t

a s

  

  

 

  

 

  

This contradicts

  H

2 . Thus k = 0. Moreover, the inequality,

0  x n ( )  z n ( )

implies

lim ( ) 0

n

x n





.

Case 2.

z n ( )  0,  z n ( )  0, 

²

z n ( )  0, 

³

z n ( )  0,

We have x(n) = z(n) - p(n)

x ( ( ))  n

, we obtain further

( ( )) ( ( )) ( ( )) ( ( ) ) x  n  z  n  p  n x  n  

 z ( ( ))  n  p ( ( )) ( ( ))  n x  n

(4)

  (1 p ( ( ))) ( ( ))  n z  n

. From equation (1) we have,

 ^{a n} ^{( )}

²

^{z n} ^{( )}  ^{q n x} ^{( ) ( ( ))} ^ ⁿ

   

  a n ^{( )} 

²

z n ^{( )}    q n ^{( )(1}  p ( ( ))) ( ( ))  n z  n

2

1

( ) ( )

( ) ,

( ) a n z n

w n n n n

z n

  



(9)

2 2

( 1) ( 1) ( ) ( )

( ) ( 1) ( )

a n z n a n z n

w n n

z n z n

 

       

                  

2 2 2

( 1) ( ( ) ( )) ( ) ( ) ( )

1 ( 1) ( ) ( 1)

w n a n z n a n z n z n

n n z n z n z n

 

    

            

2 2 2

2

( 1) ( ( ) ( )) ( ( 1))

1 ( ) ( ) ( ( 1))

w n a n z n z n

n a n

n z n z n

 

    

          

²

2 2

( 1) ( )(1 ( ( ))) ( ( )) ( )

( 1)

1 ( ) ( 1) ( 1)

w n nq n p n z n na n

n z n n a n w n

 

 

   

   

⁽¹⁰⁾

Also from Lemma (2.1) with x(n) =

 z n ( )

( ( )) ( ) , ( ) ( )

x n K n

n n

x n n

    

( ( )) ( ) ( )

z n K n

z n n

 

 



(11)

1 ( ) 1

( ) ( ( ))

K n

z n n z n



 

 

^for

 ^{( )} ⁿ   ⁿ

¹

ⁿ

².

By Lemma (2.2)

( ( )) ( ) ( ( ))

( ) ( ( ))

z n K n z n

z n n z n

  

 

 

( ) ( ) 2

K n M n

n

 



(12)

( ( )) ( ( ))

2

( ) 2

z n KM n

z n n

 _ 



Using (11) and (12) in (10)

2

2 2 2

( ) ( 1) ( )

( ) ( )(1 ( ( ))) ( 1)

2 1 ( 1) ( 1)

KM n w n na n

w n q n p n w n

n n a n

 ^  ^ ^

             

(13)

Using the inequality

2

1 , 0

4 Vx Ux V U

  U 

And put

1

₂

( )

₂

( 1), , ,

1 ( 1) ( 1)

x w n V U na n

n n a n

   

  

^{we have}

2 2

( 1) ( ) ( 1)

( 1)

( 1) ( 1) ( 1) 4 ( )

w n na n a n

n n a n w n na n

    

  

(14)

From equation (13)

(5)

2 2

( ( )) ( 1)

( ) ( )(1 ( ( )))

2 4 ( )

KM n a n

w n nq n p n

n na n

 ^  ^ ^

      

 

(15)

Summing the last inequality from

n

₂ to

n  1

we obtain

2

2 2

1

2

( ( )) ( 1)

( )(1 ( ( ))) ( )

2 4 ( )

n

n n

KM s a s

q s p s w n

sa s

 





  

    

 



Taking lim sup in the above inequality, we obtain contradiction with

H

₃. REFERENCES

[1] R. P. Agarwal, Difference and Equations and Inequalities, 2^nd Edition, Marcel Dekker, New York, 2000.

[2] R. P. Agarwal, M. Bohner, S. R. Grace and D. O. Regan, Discrete Oscillation Theory, Hindawi, New York, 2005.

[3] R. P. Agarwal, S. R. Grace and D. O. Regan, Oscillation Theory for Difference and Funtional Differential Equations, Kluwer, Dordrecht, 2000.

[4] S. H. Saker, Oscillation theory of delay differential and difference equations second and third orders, Lambert Academic Publishing (2010).

[5] R. P Agarwal, M. F. Aktas, A. Tiryaki, On oscillation criteria for third order nonlinear delay differential equations, Arch. Math (Brno), 45 (2009), 1 - 18.

[6] S. R. Grace, R. P Agarwal, R. Pavani, E. Thandapani, On the oscillation of criteria third order nonlinear functional differential equations, Appl.Math.Comp.,202(2008),102-112.

[7] S .H. Saker, Oscillation criteria of third order nonlinear delay differential equations, Mathematics Solvaca, Vol.56, no.4 (2006) pp. 433 - 450.

[8] J. Dzurina, E. Thandapani and S. Tamilvanan, Oscillatory solutions to third order half-linear netural differential equations, Elec.

J.diff. Eqns., Vol 2012 (2012), No.29, pp. 1 - 9.

[9] B. Baculikova and J. Dzurina, Oscillation of third order netural differential equations, Math. Comp. Modelling, 52(2010), 215 – 226.

[10] On the Oscillation of Third Order Linear Neutral delay Differential Equations, K V V seshagiri rao, P. V H S Sai Kumar I JMTT Vol4, Issue 11 December 2013.

[11] R. P. Agarwal, S. R. Grace and D. O. Regan, On the oscillation of certain third order difference equations, Adv. Differ. Eqn., 3(2005), 345 - 367.

[12] S. R. Grace, R. P. Agarwal and J. R. Graef, Oscillation criteria for certain third order non-linear difference equations, Appl. Anal.

Discrete Math., 3(2009), 27 - 38.

[13] J. R. Graef and E. Thandapani, Oscillatory and asymptotic behavior of solutions of third order delay difference equations, Funkcial. Ekvac.42(1999), 355 - 369.

[14] S. H. Saker, Oscillation of third order difference equations, Port. Math., 61(2004), 249-257.

[15] S. H. Saker, Oscillation asymptotic behavior of third order nonlinear neutral delay difference equations, Dyna.Sys.,15(2006)549- 568.

Some Oscillation Properties of Third Order Linear Neutral Delay Difference Equations

International Journal of Computational Engineering Research (IJCER)