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1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 1 Issue 1 ǁ Aug. 2013ǁ PP-19-21
www.ijmsi.org 19 | P a g e
On The Oscillatory Behavior of The Solutions to Second Order
Nonlinear Difference Equations
Dr. B.Selvaraj1
, Dr. P. Mohankumar2
, A.Ramesh3
1
Dean of Science and Humanities, Nehru Institute of Engineering and Technology,
Coimbatore, Tamilnadu, India.
2
Professor of Mathematics, Aaarupadaiveedu Institute of Techonology, Vinayaka Mission University,
Kanchipuram, Tamilnadu, India
3
Senior Lecturer in Mathematics, District Institute of Education and Training, Uthamacholapuram,
Salem-636 010, Tamilnadu India
ABSTRACT: In this paper we study ON THE OSCILLATORY BEHAVIOR OF THE SOLUTIONS TO
SECOND ORDER NONLINEAR DIFFERENCE EQUATIONS of form
   
   
0, 0,1,2,...
n n n n
a y p f y n N

     
KEYWORDS: Oscillatory Behavior, second order, nonlinear difference equation
I. INTRODUCTION
We are concerned with the Oscillatory behavior of the solutions of the Second order nonlinear
difference equation of the form
   
   
0, 0,1,2,...
n n n n
a y p f y n N

      (1.1)
Where the following conditions are assumed to hold.
(C1):    
,
n n
a b and  
 
n
 are positive sequence and 0
n
p  for infinitely many values of n
(C2):  
n n
  and  
lim
x
n


 
(C3):
1
1
1
n
n
s n s
R
a


  
 as n  
(C4): :
f R R
 is continuous and   0
xf x  for all 0
x  and
  0
f x
L
x
 
By Solution of equation (1.1) we mean a real sequence  
n
y satisfying (1.1) for 0,1,2,3,...
n  a
solution  
n
y is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is
called Non-Oscillatory. The forward difference operator  is defined by 1
n n n
y y y

  
Nowadays, much research is going in the study of Oscillatory behavior of the solutions of the Second order
nonlinear difference equations (refer1-9
)
II. MAIN RESULTS
In this section we present some sufficient condition of the oscillation of all the solutions of equation
(1.1)
Theorem 1
Assume the (H3) holds,   0
n

  and
 
 
 
2
2
1
(s)
1 (s)
4
s s
s
s n
a R
LR p
s n R






 

 
   
 

 
 

for 2
n n
 (1.2)
Then equation (1.1) is Oscillatory

2. On The Oscillatory Behavior of The…
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Proof:
Let  
n
y be non-oscillatory solution of equation (1.1) without loss of generality we may assume that
 
0, 0
n n
y y
  and for 1.
n n
 from equation (1.1) we have
  0
n n
a y
   for 1.
n n

Since  
n n
a y
  is non-increasing there exists a non negative constant k and 2 1.
n n
 such that
 
n n
a y k
    for 2 , 0
n n k
 
Summing the last inequality from 1 ( 1),
n to n  we obtain
1 1 1
( )
n n n n
a y a y k n n
    
Letting n   , we have
n n
a y
   , thus, there is an integer 2 , 0
n n k
 
1 1
0
n n n n
a y a y
    that is
1
n
n
y l
a
   summing the last inequality from 2 ( 1),
n to n 
We have
1
1
1
1
n
n n
s n s
y y l
a


  
This implies that n
y   as n   , which contradiction to the fact that n
y is positive, Then
  0
n n
a y
   and 0
n n
a y
 
Define
 
 
 
0,
n n
n
n
n
R a y
y




  then
 
 
   
 
1 1
n n
n n n n n
n n
R R
a y a y
y y
 
 
  
 
 
      
 
 
 
 
   
 
 
 
1 ( ) 1
1 1
( 1)
1
n n n
n n n
n n n n n
n
n n n
R R a R y y
a y a y
y y y y

  

  

 
 


 
  
 
      
 
 
 
 
   
 
 
 
1 ( ) 1
1
( 1)
1
( ( )
n n n
n n n
n n n
n
n n n
R R a R y y
p f n
y R y y

  

  
  
 



 
  
 
    
 
 
(1.3)
Consider
   
1
2 2
1
1 ;
n
n n n
n
n
y y y n n y n n



         

this implies that
   
1 1
1 n
n
y n n y
 

    (1.4)
In view of (C2),(C4) equation (1.1) and (1.4) we get from equation (1.3) that
 
 
 
 
 
2
1
1 1 1 2
( 1) 1
( )
( )
n n
n n n n
n n
n n
R y
LR p n n a R
R y

 
 
  
 
 
 
     

3. On The Oscillatory Behavior of The…
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That is
 
 
   
2 2
2 2
1 1 ( ) 1 1 ( )
( ) 2
1 ( ) 1
1 ( 1)
( ) ( ) ( )
4 4( )R
( )
n n n n n n
n n n
n n n n
a R n n R w a R
LR p
n n R n n
a R
  

  

  
 
 
  
 
     
 
 
 
This implies that
 
 
 
2
1
( )
1 ( )
4
n n
n n n
n
a R
LR p
n n R





 

 
   
 

 
 
Summing the Last inequality from 2 ( 1),
n to n 
we have
 
 
 
1
2
2
1
(s)
1 (s)
4
s s
n n s
s n
a R
LR p
s n R



 



 

 
    
 

 
 

Letting n   , we have, in view of (1.2) that n
   as n  
Which contradicts 0
n
  and the proof is complete
Example
The difference equations      
 
2
4 2 1 0; 2
n n
n y n y n

      (1.5)
Satisfies all condition of theorem 1. Here   2
n n
   and   2
f x x
 . Hence all solutions of the (1.5) are
oscillatory. In fact  
n
y =  
 
1
n
 is one such solution of the equation (1.5)
REFERENCES
[1]. R. P. Agarwal: Difference equation and inequalities- theory, methods and Applications- 2nd edition.
[2]. R.P.Agarwal, Martin Bohner, Said R.Grace, Donal O'Regan: Discreteoscillation theory-CMIA Book Series, Volume 1, ISBN :
977-5945-19-4.
[3]. B.Selvaraj and I.Mohammed Ali Jaffer :Oscillation Behavior of Certain Third order Linear Difference Equations-FarEast Journal
of Mathematical Sciences, Volume 40, Number 2, pp 169-178 (2010).
[4]. B.Selvaraj and I.Mohammed Ali Jaffer:Oscillatory Properties of Fourth OrderNeutral Delay ifference Equations-Journal of
Computer and MathematicalSciences-An Iternational ResearchJournal,Vol. 1(3), 364-373 (2010).
[5]. B. Selvaraj and I. Mohammed AliJaffer: Oscillation Behavior of CertainThird order Non-linear DifferenceEquations-
International Journal of Nonlinear Science (Accepted on September 6, (2010).
[6]. B.Selvaraj and I.Mohammed Ali Jaffer : Oscillation Theorems of Solutions ForCertain Third Order Functional Difference
Equations With Delay- Bulletin of Pure and Applied Sciences (Accepted on October 20, (2010).
[7]. E.Thandapani and B.Selvaraj: Existence and Asymptotic Behavior of Nonoscillatory Solutions of Certain NonlinearDifference
equation- Far East Journal of Mathematical Sciences 14(1), pp: 9-25 (2004).
[8]. E.Thandapani and B.Selvaraj:Oscillatory and Non-oscillatory Behavior of Fourth order Quasi-linear Difference equation -Far
East Journal of Mathematical Sciences 17(3), 287-307 (2004).
[9]. E.Thandapani and B. Selvaraj: Oscillation of Fourth order Quasi-linear Difference equation-Fasci culi Mathematici Nr, 37, 109-
119 (2007).