Oscillatory Solutions of Certain Fourth Order Nonlinear Difference Equations
B. SELVARAJ 1 , P. MOHANKUMAR 2 and A. BALASUBRAMANIAN 3
1 Dean of Science and Humanities, Nehru Institute of Engineering and Technology,
Coimbatore, Tamil Nadu, INDIA.
2 Professor of Mathematics, Department of Science and Humanities, Arupadai Veedu Institute of Technology,
Paiyanoor, Tamil Nadu, INDIA.
3 Assistant Professor in Mathematics, Arupadai Veedu Institute of Technology,
Paiyanoor, Tamil Nadu, INDIA.
(Received on: September 19, 2013) ABSTRACT
The objective of this paper is to study the oscillatory behavior of third order nonlinear neutral delay difference equation of the form
( ) ( ) 0
1
) ( 2
2 + =
∆ +
∆ n n n − k n n
n
y f q y
p
a y σ . Example is
given to illustrate the results.
Keywords: difference equations, oscillation, nonlinear.
AMS Subject Classification: 39A11.
1. INTRODUCTION
We are concerned with the oscillatory properties of all solutions of a third order nonlinear neutral delay difference equation of the form
( ) ( ) 0
1
) ( 2
2
+ =
∆ +
∆
n n n−k n nn
y f q y
p
a y
σ,(1.1)
where ∆ is the forward difference operator defined by ∆ y n = y n + 1 − y n , k is a fixed nonnegative integer and { } { } { } a n , p n , q n are real sequences with respect to the difference equation (1.1), throughout. We shall assume that the following conditions hold:
(H1) { } { } { } a n , p n , q n are real sequences and
≥ 0
a n for infinitely many values of n.
(H2) f : R → R is continuous and
( ) y > 0 , for all y ≠ 0
yf .
(H3) σ ( ) n ≥ 0 is an integer such that
( ) = ∞
∞
→ n
n σ
lim .
(H4) = ∑ − → ∞ → ∞
=
n a
R
n
n s
s
n as
1
0
.
By a solution of equation (1.1), we mean a real sequence { } y n satisfying (1.1) for n ≥ n 0 . A solution { } y n is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is called nonoscillatory. For more details on oscillatory behavior of difference equations, one can refer 1-27 .
2. MAIN RESULTS
Theorem 1: In addition to the conditions (H1), (H2), (H3), (H4), if the conditions (C1) p n ≥ 0 and ∑ ∞
=
∞
=
n
0s
p s , (C2)
∑ ∞
=
∞
=
n
0s
q s ,(C3) lim u → ∞ inf f ( ) u ≥ 0 .
Then every solution of equation (1.1) is oscillatory.
Proof:
Suppose that the equation (1.1) has non-oscillatory solution { } y n is eventually positive.Then there is a positive integer n 0 such that y σ ( n ) ≥ 0 , for n ≥ n 0 implies that
{ } y n is nonoscillatory. Without loss of
generality we can assume that there exists an integer n 1 ≥ n 0 such that
0 ,
0 ,
0 ,
0 ∆ > ≥ ∆ ≥
> n n − m n − m
n y y y
y , for
all n ≥ n 1 .
Set z n = y n + p n y n − k , then in view of (C1), 0
,
0 ∆ ≥
> n
n z
z , for all n ≥ n 1 . From equation (1.1), we have
( ) ( ) 2
2 1
n n n
n
y f q a z = − σ
∆
∆ ,
for all n ≥ n 1 . (2.1) In view of the conditions (H2), (H3), (C2) and from the equation (2.1), we obtain
∆
∆ n
n
a z 1 2
is eventually non-increasing.
We first show that 1 2 0
≥
∆
∆ n
n
a z , for
n 1
n ≥ . Suppose that, there exists an integer 0
and 1
1
2 ≥ n k >
n such that
1
1 2
k a n z n ≤ −
∆
∆ , for all n ≥ n 2 . (2.2)
Summing the inequality (2.2) from 1
2 to n −
n , we have
( 2 )
1 2
2
2 2
1
1 z k n n
z a
a n ∆ n − n ∆ n ≤ − − , for all n ≥ n 2 . (2.3) Therefore, ∆ z → −∞ n → ∞
a 1 n 2 n as
.
Then there exists an integer 0
and 2
2
3 ≥ n k >
n such that
2
1 2
k
a n ∆ z n ≤ − , for all n ≥ n 3 . (2.4)
Summing the inequality (2.4) from 1
3 to n −
n , we have
∑ −
=
−
≤
∆ 2 1
3
n
n s
s
n k a
z , for all n ≥ n 3 . (2.5)
In view of the condition (H4), and from the inequality (2.4), we obtain
∞
→
−∞
→
∆ z n as n , which is a
contradiction to the fact that ∆ z n ≥ 0 , for all large n . This shows that
1 2 0
≥
∆
∆ n
n
a z , for all large n .
Let n
n y
L = lim → ∞ . Then L is finite or infinite.
Case(i): L > 0 is finite.
In view of (H2), (H3), we have
( ) ( ) 0
lim ( ) = >
∞
→ f y n f L
n σ .
This implies that
( ) ( ) 0
2 1
)
( > f L >
y
f σ n , for all large n . Then there exists an integer n 4 ≥ n 3 and from equation (1.1), we obtain
( ) 0
2 1 1 2
2 + ≤
∆
∆ z q f L
a n n n ,
for all n ≥ n 4 . (2.6)
Summing the inequality (2.6) from 1
4 to n −
n , we have
( ) 0
2 1 1
1
2 2 14 4
4
≤
+
∆
∆
−
∆
∆ ∑
−= n
n s
s n
n n
n
L f q a z
a z
, for all n ≥ n 4 . (2.7) In view of (C2) and (C3), from inequality (2.7), we find that ∞ ≤ 0 as n → ∞ , which is a contradiction.
Case(ii): L = ∞ .
In view of (C3), there exists an integer n 4 ≥ n 3 and k 3 > 0 such that
( ) y ( ) k 3
f σ n > , for all n ≥ n 5 .
Therefore, from equation (1.1), we obtain
1 0
3 2
2 + ≤
∆
∆ z q k
a n n n , for all n ≥ n 5 . (2.8) The remaining proof is similar to that of case(i), and hence we omitted.
Thus in both cases we obtained that { } y n is oscillatory.
In fact y n < 0 , y n − m < 0 , for all large n , the proof is similar, and hence we omitted.
This completes the proof.
Corollary 1: In addition to the conditions (H1), (H2), (H3), (H4), if the conditions of Theorem1 hold. Then every bounded solution of equation (1.1) is oscillatory.
Proof: Proceeding as in the proof of
Theorem 1 with assumption that is { } y n
bounded non-oscillatory solution of equation (1.1).
Therefore, from inequality (2.6) of Theorem 1, we find that
( ) 0
2 1 1 2
2 + ≤
∆
∆ z R q f L
R a n n n
n
n , for
all n ≥ n 4 . (2.9) By the definition of R n and from the inequality (2.9), we find that
( ) L
f q R a n z n ≤ − n n
∆
∆ 1 2
, for all n ≥ n 4 . (2.10) In view of (C2), (C3) and (H4), we
have → −∞
∆
∆ n
n
a z 1 2
as n → ∞ , which is a contradiction to the fact that
1 2 0
≥
∆
∆ n
n
a z , for all large n . This shows that sequence { } y n is a bounded oscillatory solution of equation (1.1).
This completes the proof.
3. EXAMPLE
Example1: Consider the difference equation
( 1 ) 0 , 0
2 8
2
21
1 22
