International Journal of Civil Engineering and Technology (IJCIET)
Volume 10, Issue 04, April 2019, pp. 54-61. Article ID: IJCIET_10_04_007
Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=10&IType=04
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
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SOME PROPERTIES ON THE GLOBAL
BEHAVIOUR OF FIRST ORDER NEUTRAL
DELAY DIFFERENCE EQUATION WITH
POSITIVE AND NEGATIVE COEFFICIENT IN
THE NEUTRAL TERM
G. Gomathi Jawahar
Department of Mathematics, Karunya Institute of Technology and Sciences,
Coimbatore, India
ABSTRACT
In this paper some criteria for oscillatory behavior of first order Neutral Delay
Difference equation with the positive coefficient and the negative coefficient in the
neutral term is obtained, where k,l>0, {pn}, {qn} are positive sequences.
Keywords: Neutral, Delay, Difference Equations, Oscillatory, Behavior, Difference
Inequality,Positive Coefficient, Negative Coefficient
Cite this Article: G. Gomathi Jawahar, Some Properties on the Global Behaviour of
First Order Neutral Delay Difference Equation with Positive and Negative Coefficient
in the Neutral Term, International Journal of Civil Engineering and Technology, 10(4),
2019, pp. 54-61.
http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=04
1. INTRODUCTION
First 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 of
Neutral Delay Difference Equations have appeared. [3] S.S. Cheng and Y.Z. Lin (Lin, 1998)
have provided a complete characterization of oscillation solutions of first order Neutral Delay
Difference Equations with positive coefficient in the Neutral term. [6] John R.Graef,
R.Savithri, E.Thandapani (John R. Graef, 2004) have analyzed the non-oscillatory solutions
of first order Neutral Delay Differential Equations with positive coefficient in the Neutral
term. [4]Elmetwally M.Elabbasy, Taher S.Hassan, Samir H.Saker (Elmetwally M. Elabbasy,
2005) have provided oscillatory solutions of first order Neutral Delay Differential Equations
with negative coefficient in the Neutral term.
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54
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G. Gomathi Jawahar
[8]Ozkan Ocalan extensively discuss the problem of Oscillation of neutral differential
equation with positive and negative coefficients, J.Math.Anal.Appl. 33(1), 2007, 644-654.
[9]Tanaka, S. (2002) discussed the various Solutions 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.
Section I
In this section some criteria for oscillatory behavior of first order Neutral Delay Difference
Equation,
( xn + pn xn−k ) + qn f ( xn−l ) = 0
(1.1)
Is obtained where k,l > 0 , {pn}, {qn} are positive sequences.
The following assumptions has been made to prove the results:
H1: f (u) is an increasing function and u f (u) > 0
H2: There exists a function w such that w (u) > 0 for u > 0 and w (u) f (v) > f(u v).
H3: φ (u) is an increasing function such that u φ(u) > 0 and |φ(u+v)|<|f(u)+f(v)|
2. EXISTENCE OF OSCILLATORY SOLUTIONS
Theorem 1.1 Every solution of the equation (1.1) is oscillatory if there exists a constant λn such
that 0 < λn < 1 for some n ≥ n0 and the difference inequality
Δzn+Qn φ(zn-l+k) ≤ 0
Satisfies where
(
(1.2)
)


1− 
q
n−k n−l 
Q = min  q ,

n
n n
wp


n−l
z =
n
and
(
m
 Qn  yn − l
n +k
0
)
Proof
Suppose to the contrary that there is a non-oscillatory solution {xn} of (1.1). Suppose that
xn > 0 for all n ≥ n0.
Let yn= xn + pn xn-k.
Then by the equation (1.1),
Δyn = -qnf(xn-l) < 0
yn+1 < yn, yn is a decreasing function, yn+1-yn = -qn f (xn-l)
yn+1+ qn f (xn-l) = yn
yn > qn f (xn-l) for some n ≥ n0, taking summation from n0 to m, m > n0
(
m
m
 y n   q n f xn − l
n=n
n=n
0
0

)
(

(
(
)(
) (
m
 n q n f x n − l + 1 − n q n f x n − l
n=n
0
)
(
)
(
m
m
1− 
q
f x
 Qn f xn − l +

n−k n−k
n−k −l
n=n
n=n +k
0
0
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55
))
)
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Some Properties on the Global Behaviour of First Order Neutral Delay Difference Equation with
Positive and Negative Coefficient in the Neutral Term

(
)
(
)(
m
m
Q wp
f x
 Qn f xn − l +

n
n−l
n−k −l
n=n
n=n +k
0
0

(
)
(
m
m
Q f p
x
 Qn f xn − l +

n
n−l n−k −l
n=n
n=n +k
0
0

(
(
m
Q f x
+f p
x

n
n−l
n−l n−k −l
n=n +k
0

(
m
 Qn  xn − l + pn − l xn − k − l
n +k
0
=

m
 Qn yn − l
n +k
0

)
)
))
)


m
 Qn  yn − l  0
n +k
0
Let
z =
n
(
m
Q  y

n
n−l
n=n +k
0
)
Also, Δzn = zn+1 - zn
=

(


m
 Qn + 1  yn + 1 − l − Qn  yn − l
n +k
0
)






+Q
=Q
 y
+Q
 y
 y
+ ....... +
n + k + 1  n + k + 1 − l 
n + k + 2  n + k + 2 − l  n + k + 3  n + k − l 
0
0
0
 0

 0

 0

(
)




 −Q
Q
y
−Q
 y
 y
m +1 m +1− l
n + k  n + k +1
n + k + 1  n + k + 1 − l 
0
0
 0

 0




−Q
 y
+ .... − Q  y
n + k + 2  n + k − l + 2 
m
m−l
0
 0






=Q
y
−Q
 y
m +1
m +1− l
n + k  n + k − l 
0
 0




z  − Q
  y
n
n +k
n + k −l 
0
 0

> -Qn φ (yn-l), n≥ n0+k
> -Qn φ (zn-l+k)
Δ zn + Qn φ (zn-l+k) > 0
(1.3)
Condition (1.3) holds when zn is eventually positive solution. Contradiction to the equation
(1.2). Since
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56
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G. Gomathi Jawahar
z =
n
(
m
 Qn  yn − l
n +k
0
)
and
y =x + p x
n n
n n−k
,
Hence we say that {xn }is an oscillatory solution of the equation (1.1).
Theorem 1.2
Δyn + qn f
Every solution of the equation (1.1) is oscillatory, if it satisfies the condition
(1 − n ) P }, and
(xn-l ) ≤ 0, where yn = xn + pn f(xn-k) and Qn = min{ λn pn ,
if there exists
n
a function  such that  (u ) > 0, for u > 0 and f(u)=u,
f (uv )
 n −l
≤
 (u ) f (v ) and f (uv) ≥ uv,
λ n is positive constant.
Proof
Suppose {xn} is a non-oscillatory solution. Let xn > 0 and let us assume {xn} is eventually
positive. Let yn = xn + pn f(xn-k)
yn = xn + pn f(xn-k)
> pn f(xn-k)
Hence yn > pn f(xn-k)
= λn pn f(xn-k) + (1-λn ) pn f(xn-k)
> Qn f(xn-k) + Qn f(xn-k)  n-l+k
> Qn f(xn-k) + Qn f(xn-k xn-l+k)
> Qn f(xn-k) + xn-k xn-l+k > 0
Since yn > 0, Δyn > 0
i.e, Δyn + qn f(xn-l) > 0, yn > 0
This contradicts our assumption. Hence {xn} is an oscillatory solution.
Theorem 1.3
If lim ∑qn > 0, and {xn}is an eventually positive solution of
n →
Δ(xn + pn xn-k) + qn f(xn-l) = 0,
then Δzn +
Mqn
pn
2
(1.4)
( p n − 1) zn+k-l ≤ 0
Where zn = xn+ pn xn-k
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57
(1.5)
editor@iaeme.com
Some Properties on the Global Behaviour of First Order Neutral Delay Difference Equation with
Positive and Negative Coefficient in the Neutral Term
Proof:
Equation (1.4) becomes,
Δzn + qn f(xn-l) = 0
Δzn = -qn f(xn-l) < 0
zn+1-zn < 0
zn+1 < zn
Hence zn is decreasing.
From the equation (1.5) ,
pn xn-k = zn - xn
(1.6)
zn+k = xn+k + pn xn
(1.7)
From the equation (1.5),
Since zn is decreasing
zn > zn+k ≥ pn xn
From the equation(1.6),
pn2 xn-k= pn zn - pn xn (multiplying by pn)
(1.8)
pn xn = zn+k - xn+k
p x z
n n
n+k
- pn xn ≥ -zn+k
Substituting inthe equation (1.8) ,
pn2 xn-k = pn zn - pn xn
pn2 xn-k ≥ pn zn – zn+k
(1.9)
pn2 xn-k ≥ pn zn+k - zn+k
xn-k ≥
xn-l ≥
pn − 1
pn
2
pn − 1
pn
2
zn+k
(2.0)
zn+k-l
(2.1)
Δzn + qn f(xn-l) = 0
= > Δzn = -qn f(xn-l)
Δzn ≤ -qn M
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pn − 1
pn
58
2
zn+k-l
editor@iaeme.com
G. Gomathi Jawahar
Δzn + M
q n ( p n − 1)
pn
zn+k-l ≤ 0
2
(2.2)
Hence the theorem.
Section II
In this section some criteria for oscillatory behavior of first order Neutral Delay Difference
Equation,
(
)
 x −q x
+p x
=0
n n n−k
n n−l
(2.3)
is obtained where k,l > 0 and {pn} and {qn} are positive sequences.
3. SOME OSCILLATORY RESULTS
Theorem 2.1
Every solution of the equation (2.3) is oscillatory if k, l, p n , q n > 0
Proof
Case 1
Suppose { x n }is a non oscillatory solution of equation (2.3) ,
let x n >0 and { xn } is eventually positive, xn −1 < x n
(
)
 x −q x
=− p x
n n n−k
n n−l
(
= −p z
+q
x
n n−l
n−l n−k −l
z + p z
+p q
x
=0
Hence n n n − l n n − l n − k − l
z
n +1
z
(
n +1
(
)
−z + p z
+q
x
=0
n
n n−l n−l n−k −l
p
p
)
(
)
+p z
+q
x
=z
n n−l n−l n−k −l
n
(
)
z
+q
x
0
n n−l n−l n−k −l
)
x
−q
x
+q
x
0
n n−l
n−l n−k −l
n−l n−k −l
p x
0
n n−l
Which is a contradiction. Since p n is positive and x n −l is eventually positive.
Hence equation (2.3) has a oscillatory solution.
Case 2
Let
(
us
assume
)
xn
<
0
and
{ x n }is
eventually
negative,
then
x
n −1
x
 x −q x
=− p x
n
n n−k
n n−l
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59
editor@iaeme.com
n
Some Properties on the Global Behaviour of First Order Neutral Delay Difference Equation with
Positive and Negative Coefficient in the Neutral Term
=
(
−p z
+q
x
n n−l n−l n−k −l
)
z + p z
+p q
x
=0
n
n n−l
n n−l n−k −l
z
n +1
−z + p z
+p q
x
=0
n
n n−l
n n−l n−k −l
p
p
(
(
)
z
+q
x
0
n n−l n−l n−k −l
)
x
−q
x
+q
x
0
n n−l n−l n−k −l n−l n−k −l
p x
0
n n−l
Which is a contradiction, since p n is positive and x n −l is eventually negative solution.
Hence equation (2.3) has an oscillatory solution.
4. Some Non oscillatory results
Theorem 2.2
(
)
Let {xn} be an eventually positive solution of  x − q x
+p x
=0 ,
n n n−k
n n−l
xn > 0 and
let z n = x n -q n x n− k
(2.4)
Assume k,l > 0, p n ,q n > 0. Then z n is eventually non increasing positive function.
Proof
From the equation (2.3),
Δz n = -p n x n− l < 0
z n +1 - z n < 0
z n +1 < z n
Hence z n is eventually non increasing positive function.
Theorem 2.3
Every non oscillatory solution of the equation (2.3) converges to zero monotonically for large
n as n→∞ if k, l > 0, q n > 0, p n > 0.
Proof
Suppose {x n }is a non-oscillatory solution of (2.3). Let us assume {x n } is eventually positive
(if x n is eventually negative, the proof is similar).
Hence from the theorem 2.2, z n is eventually non increasing positive function.
From equation (2.4),
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G. Gomathi Jawahar
xn ≥ zn
Here
zn = α ≥ 0
lim x n ≥ lim
n →
n →
If α > 0, then from the equation (2.3),
pn xn−l < 0
z n →-∞, which contradicts that z n is positive function.
Δz n = -
Hence
lim
n →
Hence we take α = 0, therefore
lim
x n = 0.
n→
Hence every non oscillatory solution of the equation (2.3) converges to zero monotonically
as n→∞.
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[2]
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[5]
[6]
[7]
[8]
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D.D Bainov and D.P Mishev, Oscillation Theory for Neutral Differential Equations with
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S. S. Cheng and Y. Z. Lin(1998), Complete Characterization of an oscillatory Neutral
Difference Equations. J. Math. Anal. Appl. 221 , 73-91.
Elmetwally M. Elabbasy, T. S. (2005),Oscillation criteria for first-order Nonlinear Neutral
Delay Differential Equations, Electric Journal of Qualitative theory of differential
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I.Gyori and G.Ladas, Oscillation Theory of Delay Differential Equations With
Applications, Clarendon Press, Oxford,1991.
John R Graef, R. Savithiri , E. Thandapani, (John R Graef, 2004) Oscillation of First order
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Ozkan Ocalan, Oscillation of neutral differential equation with positive and negative
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Equations, Hiloshima Mathematical Journal(32) , 79-85.
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