This document investigates the oscillatory behavior of second order neutral delay difference equations. It presents two theorems on the oscillation of solutions to such equations. Theorem 1 states that if the sum of coefficients diverges, then every solution is oscillatory. Theorem 2 provides conditions under which the sum of terms of a non-oscillatory solution is bounded. An example equation is also presented and shown to satisfy the conditions of Theorem 1, implying all its solutions oscillate. The document contributes to the study of oscillation properties of neutral delay difference equations.

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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 02, February 2019, pp. 150–154, Article ID: IJMET_10_02_019
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=2
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication Scopus Indexed
OSCILLATION BEHAVIOUR OF SECOND
ORDER NEUTRAL DELAY DIFFERENCE
EQUATIONS
G. Gomathi Jawahar
Department of Mathematics
Karunya Institute of Technology and Sciences,
Coimbatore, Tamilnadu, India
ABSTRACT
The aim of this paper is to investigate some properties of oscillation of Neutral
Delay Difference Equation of the second order of the form,
)) - ) = 0 , for
. Example is provided to prove the results.
Key words: Neutral, Delay, Oscillation, Non Oscillation.
Cite this Article: G. Gomathi Jawahar, Oscillation Behaviour of Second Order
Neutral Delay Difference Equations, International Journal of Mechanical Engineering
and Technology 10(2), 2019, pp. 150–154.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=2
1. INTRODUCTION
Second order neutral delay difference equation is gaining interest because they are the discrete
analogue of differential equations. In recent years, several papers on oscillation of solutions
neutral delay difference equation have appeared . [1] Ozkan Ocalan extensively discusses the
problem of oscillation of neutral differential equation with positive and negative
coefficients,J.Math.Anal.Appl.33(1),2007,644-654.[4]Tanaka, s. (2002) discussed the various
solution of Oscillation First order Neutral Delay Differential Equations. Here some oscillation
results in difference equations based on the existence results of differential equations are
provided. Examples are provided to illustrate the results.
2. SOME OSCILLATORY RESULTS
2.1. Theorem
If ∑ ) , then every solution of the equation
)) - ) = 0, is oscillatory, forsome
suppose be a non oscillatory solution of,
)) - ) = 0 (1.1)

2. Oscillation Behaviour of Second Order Neutral Delay Difference Equations
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without loss of generality. let us assume that is eventually positive
solution,
let
Obviously,
)) - ) = )) - ) (1.2)
) - ) - ) (1.3)
Define
)
)
, then
Equation (1.3) becomes
.
)
- .
)
- )
.
)
+
)
)
.
) )
)
.
)
)
. )
)
)
) )
)
Generalizing,
∑ )
When s ,∑ )
Hence the proof

3. G. Gomathi Jawahar
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2.2. Theorem
let be a non oscillatory solution of the equation (1.1) and if the following assumptions
holds,
( ) :
)
, 0
( ) : ∑ ( )) then,
∑ ∑ * ( )+
Where , , n, m,
Proof
let be a non oscillatory solution of the equation (1.1) and without loss of generality,
assume that 0.
Let
)) - )
)) )
From ( ) : ) - ) - )
Since
) - ) - ) (1.4)
Define , , ,
substituting value in equation(1.4),we have
( ) ( ) ( )
)
( )
) ( ) ( )
( )
) ( ) ( )

4. Oscillation Behaviour of Second Order Neutral Delay Difference Equations
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( )>-
) ( ) ( )
) ( ) ( )
) ( ) ( )
[ ) ) ( )
]
taking summation from s= to n,
∑ ∑ ( ))
∑ [ ) ( )]
∑
∑ [ ) ( )] ∑
∑ [ ) ( )] ∑
∑ [ ) ( )] ∑
This complete the proof.
3. EXAMPLE
3.1. Consider the second order neutral delay difference equations
)) + ) (1.5)
Here
)) )
Equation (1.5) satisfies all conditions of theorem (1.1.1) and hence all its solutions are
oscillatory. one such solution is ) .

5. G. Gomathi Jawahar
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REFERENCES
[1] Ozkan ocalan, oscillation of neutral differential equation with positive and Negative
coefficients,j.Math.Anal.Appl.331(1)(2007),644-654.
[2] John R.Graef, R.s.(2004). Ocillation of First order Neutral Delay
[3] Difference Equations, Electric Journal of Qualitative of differential
equations,proc.7thcoll.QTDE,NO,12,1-11.
[4] Tanaka, S(2002), Oscillation of solution of First order Neutral Delay Differential
Equations,Hiloshima Mathematical Journal,32,73-85.
[5] S. S. Cheng and Y. Z. Lin (1998), ‘Complete Characterization of an oscillatory Neutral
Difference Equations’. Math. Anal. Appl. 221, 73-91.
[6] John R.Graef, R.Savithri, E.Thandapani,’Oscillation of First order Neutral Delay
Differential Equations’, Electronic Journal of Qualitative theory of differential Equation,
Proc. 7th coll. QTDE, No. 12 , (John R. Graef, 2004), 1-11.