OSCILLATION PROPERTIES OF CERTAIN TYPES OF FIRST ORDER NEUTRAL DELAY DIFFERENCE EQUATIONS

G.GOMATHI JAWAHAR


ABSTRACT


In this paper some sufficient

were obtained with the assumption of e the following conditions.

condition for the oscillation of first order


neutral delay difference equation were


H1:


{ pn } is an positive sequence.



obtained. .


KEYWORDS :

Neutral Delay Difference Equation, Oscillation, Nonoscillation, Eventually

H2: f is a continuous function such that


uf (u) 0 .


H3: If there exists a function w such that


positive.

w(u)>0, for u>0 and

image

image

f (uv) w(u) f (v) .



Introduction 1.1

H4: If there exists a function φ such that



In this paper some sufficient

φ(u) is increasing and

uφ(u) > 0, for

u 0


condition for the oscillation of first order

image

image

& φ(u + v)

image

image

f (u) f (v) .


neutral delay difference equation of the form


(an xn pn xnk ) + qn f ( xnl ) = 0, n N (n0 )

(1.1.1)


and


( xn + pn xnk ) + qn f ( xnl ) = 0, n N (n0 )

(1.1.2)


1.2 Existence of Oscillatory Solutions


In this section, I obtain some sufficient condition for the oscillatory solutions of the equation (1.1.1) and (1.1.2)

Theorem 1. 2.1

Assume that


image

pn 1


Hence

zn −∞ ,


n → ∞

ank


and xn be an eventually positive solution of


as n .


.Which contradicts the fact that



the equation (1.1.1) and

zn is eventually positive. Hence the proof.



yn = (an xn pn xnk )


. Then eventually yn >0.

Theorem 1.2.2


pn


Proof

Assume that pn, qn > 0 and


ank

image

1.



Let us consider xn > 0, xn-l>0, xn-k >0 for


lim

image

inf q (1+ pnl λnk ) > 0,


some n > n1.


If,

n→∞ n


q= qn

qnk

From the equation (1.1.1),

where

image

a

n

nl

,then every solution of



yn


= qn f (xnl ) < 0.


Hence yn is a

equation (1.1.1) is an oscillatory solution.


decreasing function.


Suppose yn is not eventually positive, then eventually yn < 0.


Hence there exists n2 > n1 and M>0, such

Proof


Let us assume the contradiction that equation (1.1.1) has an non oscillatory solution. Let us consider xn is eventually


that


Let,


Then,


yn < −M .


zn = an xn > 0.


zn = yn + pn xnk .


positive.


Let us consider xn > 0, xn-l>0, xn-k >0 for some n > n1.


By theorem 1.2.1, yn is eventually positive.


zn < −M +

image

pn ank


.

znk


Also we have

yn

= qn f (xnl )


n

n

λ q +

p λ q

nl nk n

image

q


yn = an xn pn xnk

nk



pnl λnk


yn ≤ −qn xnl


(1.2.1)

Hence

limn→∞ inf qn (1 +


qnk

image

) λn ,



yn


qn


ynl + pnl xnk l

image

a


Therefore,


lim


n→∞

image

n

inf q (1 + pnl λnk ) 0.

qnk

nl



yn


image

= qn ynl

anl


image

(1.2.2)

n n a


nl

anl

qnk


has oscillatory solution where,


q = qn


(1 λ )q

image

a

n

nl

, Hence we have,

Qn = minλn qn ,

nk

wpnl

nl


Proof


Suppose to the contrary that there is a non oscillatory solution xn. Assume that ,

m

yn

n=n0

m

> Qn

n=n0


f ( x


nl ) +

m

Qn

n=n0 +k


f ( p


nl


xnk l )


xn > 0,

m

yn

n=n0

m

>

n=n0


Qn { f ( x


nl


) + f ( p


nl


xnk l )}


For all n>n0. Let

m m

yn > Qn {φ( xnl + pnl xnk l )}

yn = xn + pn xnk

n=n0

n=n0 +k



.

( yn ) = qn f (xnl ) < 0

m

yn >

n=n0

m

Qn {ynl }

n=n0 +k


Also yn+1<yn, yn is decreasing function.,


Let


m

zn = Qnφ{ynl } > 0.

Hence

yn+1 + qn f ( xnl ) = yn

n0 +k



yn > qn f ( xnl ), n n0.


Then

zn = zn+1 zn



Taking summation from n0 to m , m>n0,


zn =

m

(Qn+1φ{yn+1l } Qnφ( ynl ))

n=n0 +k


m m

yn >

qn f ( xnl )

zn

= Qm+1φ( y


m+1l

0

) φn +k

φ( y

n0 +k l )

n=n0

n=n0


zn > −Qnφ( ynl )


m m

yn > ((λn qn f ( xnl ) + (1 λn )qn f ( xnl ))

zn > −Qnφ(znl +k )

n=n0


m

n=n0


m m


zn + Qnφ(znl +k ) > 0.


This condition holds

yn > Qn f ( xnl ) +

(1 λnk )qnk f ( xnk l )

n=n0

n=n0

n=n0 +k

when zn

is eventually positive solution. This



m m m

is a contradiction to the equation (1.2.2).

yn > Qn f ( xnl ) +

Qn w( pnl ) f ( xnk l )

n=n0

n=n0

n=n0 +k

Hence the proof compltes. Similarly we prove that, when xn is eventually negative.


2.1 Examples


Example 2.1.1


Consider the first order neutral delay difference equation

References


[1] R.P.Agarwal , ‘Difference Equations and Inequalities’, Marcel Dekker,New York,(1992).


[2] S.S.Cheng and W. Patula, ‘An Existence theorem for a Nonlinear Difference Equations’, Nonlinear Anal.20, 193-203 (1992).


[3] D.A.Georgiou,E.A.Grove and G.Ladas, ‘

Oscillations of Neutral Difference Equations

,Appl.Anal.33,243-253(1989).

(nxn


Here

xn1

) + (2n + 3)x


n2

3 = 0, n > 0

[4] S. R. Grace, Giza, H. A. El-Morshedy ,

On the Oscillation of Certain Difference Equations’, Mathematica Bohemica, 125 ,No. 4,

421-430(2000).


an = n, k = 1,l = 2, pn = 1, qn = (2n + 3)


All the conditions of the theorem 1.2.2 are satisfied. Hence all its solutions are oscillatory. One such solution is (-1) n.

Example 2.1.2


Consider the first order neutral delay difference equation


[5]G.Ladas, ‘ Recent Developments in the Oscillation of Difference Equations’, J.Math.Anal.Appl.153,276-287(1990).


[6]G.Ladas, ch.G.Philos and Y.G.sficas, ‘Necessary and Sufficient Conditions for the Oscillations of Difference Equations’, Liberta Math.9 ,121-125(1989).


[9] Ozkan A Ocalan and A Omer Akin, ‘ Oscillation Properties for Advanced Difference Equations’, Novi Sad J. Math,Vol. 37, No. 1, 39-47(2007).


[10] E. Thandapani and P. Mohan Kumar , ‘


(xn


Here

image

1

n 1


xn1 ) +

2n + 3

image

(n 2)3


xn2


3 = 0, n > 2

Oscillation and Non Oscillation of Nonlinear

Neutral Delay Difference Equations’, Tamkang

Journal of Mathematics,Volume 38, Number 4,

323-333,( 2007).


[11] Willie E. Taylor, Jr. and Minghua Sun, ‘


an = 1, k = 1,l = 2, pn

image

= − 1

n 1


, qn


image

= 2n + 3 (n 2)3

Oscillation Properties of Nonlinear Difference Equations’, Portugaliae Mathematica,Vol. 52 Fasc. 1( 1995).


Hence all the conditions of the theorem 1.2.2 are satisfied.

Hence all its solutions are oscillatory. One such solution is n(-1)n.