 Oscillation behavior of certain fourth order neutraldifference equations

(1)

Oscillation behavior of certain fourth order neutral difference equations

B. SELVARAJ and J. DAPHY LOUIS LOVENIA Department of mathematics

Karunya University, Karunya Nagar, Coimbatore-641 114, Tamil Nadu, India.

e-mail: [email protected], [email protected]

ABSTRACT

In this paper, we establish the sufficient conditions for oscillation of fourth order neutral difference equation of the form

0 ))

(

( ² ₃

2    

 r_n y_n p_ny_n__k q_ny_n__l_ by using comparison method. Example is provided to illustrate the results.

Keywords : Oscillation, Neutral difference equations, comparison method.

AMS Classification :39 A 11

1. INTRODUCTION

Consider the following fourth order non linear neutral difference equation of the form

0 ))

(

( ² ₃

2    

 r_n y_n p_ny_n__k q_ny_n__l_

(1.1) where nN(n₀){n₀,n₀ 1,n₀ 2,...}, ,...},n0 is a non-negative integer and  is the forward difference operator defined by y_n  y_n_₁  y_n_.. T he follow ing conditions are assumed to hold:

(a) {p_n}^and{q_n} are non-negativee

real sequences with {q_n} not identically zero for infinitely many values of n.

(b) There is a positive constant p such that 0 p_n  p1 and k and l are positive integers.

(c) {r_n}is a positive sequence of real numbers for n N(n₀) such

that











0

.

n n rn

n

Let  max{k,l}. By a solution

(2)

of (1.1) we mean a real sequence {y_n}

which is defined for all n N₀ ^and

satisfies equation (1.1) for all n N )

(n₀

N . As it is customary, a solution

}

{y_n of (1.1) is said to be oscillatory if the terms yn of the sequence are not eventually positive or not eventually negative; otherwise it is called non- oscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

In recent years, there has been an increasing interest in the study of oscillatory and asymptotic behavior of solutions of fourth order difference equations^2-20 and references cited therein.

Following this trend, in this paper we obtain some sufficient conditions for the oscillation of all the solutions of equation (1.1) using comparison method. An example is provided to illustrate the result.

2. Main results

Let {xn} be a real sequence. We define a sequence {zn} by z_n  y_n p_n

k

yn_ ^, ^(2.1)

) (n₀ N

n  where {pn} and k are as defined as earlier.

In this section we present some sufficient conditions for the oscillation of all solutions of equation (1.1).

Lemma2.1 Let {yn} be a positive

solution of the equation (1.1), then there are only the following two cases for large n,

, 0 ,

0 ,

0 )

(I y_n  z_n  z_n  r_n²z_n  .

0 ) ( ² 

 r_n z_n

, 0 ,

0 ,

0 )

(II y_n  z_n  z_n  r_n²z_n  0

) ( ² 

 r_n z_n

The proof follows from Discrete Kneser’s Theorem [1,Theorem 1.7.11]

Lemma 2.2 Let {yn} be a positive solution of the equation (1.1) and not ide nt ic a lly ze ro. T he n for eve ry

1 0

, 

 , there exists a large positivee integer N such that for all n  N_^,

(3) 2

( ),

n 3! n n

y  n r y

   w he re n⁽³⁾  n )

2 )(

1 (n n

n ^.

Proof. The proof can be found in [16] and [20].

Lemma2.3 If {yn}

is an eventually positive solution of (1.1) , then there exists an integer n N(n₀)_{such that}

n n n

n z y z

p  

 )

1

( ^forn N^.

Proof. From definition of z_n, we have z n yn_forn N_{. Hence}

n n n

k n n n n

z p y

z p z y

y p z y

) 1 ( 













for n N^.

(3)

Theorem 3.1 Assume that there exists a constant 0 ₀ 1, such that t he f irs t orde r de lay dif f e re nc e

equation ( 3)

! 3

) 3 0 (







y_n  q_n n l 0 ]

1

[  p_n__l_₃ y_n__l_₃  _{(3 .1 )}

is oscillatory. Then equation (1.1) is oscillatory.

Proof: Let {yn} be an eventually positive solution of (1.1) and define

k n n n

n y p y

z   _ ^{(3 .2 )}

Then from hypotheses (a) and (b) there exists an integer n ₁ n₀ such that

n 0

y  and 0 ) ( ²

2  

 r_n z_n for all n n₁.^{By Lemma} 2.1, there exists for some large n ₂ n₁, such that either

, 0 ,

0 ,

0    ² 

 _n _n _n _n

n z z r z

y

. 0 ) (

, r_n²z_n 

, 0 ,

0 ,

0    ² 

 _n _n _n _n

n z z r z

y OR

0 ) ( ² 

 r_n z_n

Hence lim_n__ z_n 0. Therefore by Lemma 2.2 for every ,01^{, there}^e exists a N_ such that for all n  N_^,

),

! ( 3

2 ) 3 (

n n

n n r z

z    

(3 .3 ) From the equation (2.1)

3 3 3

3   



_l  _n _l  _n _l _n _l

n z p y

y

and we have

0 ] [

)

( ² ₃ ₃ ₃

2    

 r_n z_n q_n z_n__l_ p_n__l_ y_n__l_ for all large n.

Since z _n y_n^andz_n ⁰, we obtain

0 ]

1 [ )

( ² ₃ ₃

2    

 r_n z_n q_n p_n__l_ z_n__l_

for all sufficiently large n. Using (3.3), for every 0 1,

) 3

! ( 3

! ) 3

( ² ⁽³⁾

2     

 r_n z_n   q_n n l

[1 p_n__l_₃][r_n__l_₃(²z_n__l_₃)]0.

for all large n.

Let v_n (r_n²z_n)^{. Thus}{v_n} satisfies

) 3

! ( 3

) 3

 (





v_n  q_n n l 0

] 1

[  p_n__l_₃ v_n__l_₃  , f or n la rge enough and

for every 0 1.

Using result from [1], the difference equation

] 1

[ ) 3

! (

3 ³

) 3

( 







w_n  q_n n l p_n_l_ w .

3  0



l

wn

(4)

has an eventually positive solution for every 0 1. This contradicts the fact that (3.1) is oscillatory. When {yn} is eventually negative solution, {-yn} will be an eventually positive solution.

Proof of Theorem 3.1 is complete.

3. Example: Consider the difference

equation ² ² ² ²

1 1

1 9

[ ( 1) ( )] 8(2 6 5) (9 21 14) 0, 1

2 2

n n n n n

n n y y _ n n n n y _ n

           

2 2 2 2

1 4 1

1 9

[ ( 1) ( )] 8(2 6 5) (9 21 14) 0, 1

2 2

n n n n n

n n y y _  n n _ n n y _ n

            

 

1 1

[ (n n 1) (y_n y_n_ )] 8(2n 6n 5) (9n 21n 14)y_n_ 0,n 1

            

 

(4 .1 )

Here , 8(2 6 5)

2

1 ₂







 q n n

p _n

n n

), 1 ( ),

14 21 9

2 (

9 2

4    

 n n r_n n n

n

. 2 , 1 

 l

k It is easy to see that all solutions of equation (4.1) are oscillatory.

In fact ^{yn^}^{

 

¹ⁿ^} is one such solution of equation (4.1)

REFERENCES

1. R.P. Agarwal, M. Bohner, S.R. Grace and D.O. Regan, Discrete Oscillation Theory, Hindawi Publishing Corpo- ration (2005).

2. R.P. Agarwal and S.R. Grace, Osci- llation of higher order difference equations, Appl. Math. Lett., 13,

81-88 (200).

3. R.P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York (2000).

4. R.P. Agarwal, E. Thandapani and P.J.Y. Wong, Oscillation of higher order neutral difference equations, Appl. Math. Lett., 10(1), 71-78 (1997).

5. S.S. Chang, on a class of fourth order linear reccurence equations, Internat. J. Math & Math. Sci. 7, 131-149 (1984).

6. J. Popenda and E. Schmeidal, On the solution of fourth order diffe- rence equations, Rocky Mountain.

J. Math. 25, 1485-1499 (1995).

7. B. Selvaraj, M. Mallika Arjunan and V. Kavitha–‘Existence of Solutions for Impulsive Nonlinear Differential Equations with Nonlocal Conditions’

–J KSIAM, Vol. 13, No. 3, September pp. 203-215 (2009).

8. B. Selvaraj and J. Daphy Louis Lovenia – ‘Oscillation Behavior of Fourth Order Neutral Difference Equations with Variable Coefficients’

–Far East Journal of Mathematical Sciences (FJMS), Vol. 35, Issue 2, pp. 225-231 (2009).

9. B. Selvaraj and I. Mohammed Ali Jaffer – ‘Oscillation Behavior of Certain Third Order Linear Difference Equations’, accepted for publication in “Far East Journal of Mathema- tical Sciences” in February 5, (2010).

10. B. Selvaraj and G. Gomathi Jawahar – ‘Asymptotic Behavior of Fourth Order Non-linear Delay Difference Equation’, accepted for publication in “Far East Journal of Mathema-

(5)

tical Sciences” in April 22, (2010).

11. B. Selvaraj and I. Mohammed Ali Jaffer – ‘Oscillatory Properties of Fourth Order Neutral Delay Difference Equations’, accepted for publication in “Journal of Computer And Mathe- matical Sciences” in May 4, (2010).

12. E. Thandapani and B. Selvaraj –

‘Oscillatory Behavior of Solutions of Three Dimensional Delay Diffe- rence Systems’ – Radovi Mate Maticki, Vol. 13, 39-52 (2004).

‘Existence and Asymptotic Behavior of Non Oscilatory Solutions of Ce rtain Nonline a r Differe nc e Equations’ – Far East Journal of Mathematical Sciences (FJMS), 14 (1), 9 – 25 (2004).

‘Oscillatory and Non Oscillatory Behavior of Fourth Order Quasilinear Difference Equations’ – Fast East Journal of Mathematical Sciences (FJMS), 17 (3), 287-307 (2004).

‘Behavior of Oscillatory and Non Oscillatory Solutions of Certain Fourth Order Qualsilinear Difference

Equations’ – The Mathematics Education, Vol. XXXIX (4) 214- 232 (2005).

‘Oscillation of Fourth Order Quasi- linear Difference Equations ‘ Fasciculi Mathematic Nr 37,109-119 (2007).

17. E.Thandapani and I.M. Arockiasamy, Oscillatory and asymptotic properties of solutions of non linear fourth order difference equations, Glasnik M a t hem a t ik i 37 (5 7 ),1 21 -1 3 3 (2002).

18. E.Thandapani and I.M. Arockiasamy, Fourth order nonlinear oscillations of difference equations, Compu- ters and Mathematics with Appli- cations, 42, 357-368 (2001).

19. E. Thandapani and I.M. Arockia- samy, Oscillatory and asymptotic behaviour of fourth order non linear neutral delay difference equations Indian J. Pure Appl, Math., 32(1), 109-123 (2001).

20. E. Thandapani, P. Sundaram and B.S. Lalli, Oscillation theorems for higher order nonlinear delay diffe- rence equations, Computers Math.

Appl., 32(3), 111-117 (1996).