International Journal of Scientific & Engineering Research, Volume 2, Issue 2, February-2011 1
ISSN 2229-5518
B. Selvaraj1, I. Mohammed Ali Jaffer2
1The Dean of Science and Humanities, Nehru Institute of Engineering and Technology
2Research Scholar, Department of Mathematics, Karunya University, Coimbatore, Tamil Nadu, India
Key words: Linear, Nonlinear, Difference equations, Oscillations and Non-oscillation.
—————————— • ——————————
1 Introduction
We are concerned with the oscillatory properties of all solutions of third order nonlinear difference equations of the form
2 qn x
+ c x
+ p x
+ q f ( x
) = 0; n = 0,1, 2,...
(1.1)
\ an
n n n cr
)
n n n n +1
2 qn ( x ) x
+ c x
+ p x
+ q f ( x
) = 0; n = 0,1, 2,...
(1.2)
\ an
n n n n cr
)
n n n n +1
2 qn x
+ c x
+ q f ( x
) = 0; n = 0,1, 2,...
(1.3)
2 qn ( x ) x
\ an
+ c x
n n n cr
)
+ p x
n n +1
+ q f ( x
) = 0; n = 0,1, 2,...
(1.4)
\ an
n n n n cr
)
n n n n +1
Where the following conditions are assumed to hold.
(H1) {an },{ pn },{qn } and {cn } are real positive sequence and qn 0
for infinitely many values of n .
(H2)
f : R � R is continues and
xf ( x) > 0 for all x 0 .
(H3) there exists a real valued function g such that
f (un ) f (vn ) = g (un , vn )[(un + cnun cr ) (vn + cn vn cr )], for all un 0, vn 0, c 0, n > cr > 0 and
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ISSN 2229-5518
g (un , vn )
L > 0 E R.
(H4) : R � R is continues for all x
0, ( xn ) > 0 .
(H5)
(n + 1) pn .
n= M
(H6)
q 2
n .
n= 0 an
(H7) (n + 1)qn = .
n= 0
(H8) n
= .
n= 0 nqn
By a solution of equation (1.1) –(1.4), we mean a real sequence {xn } satisfying (1.1)-(1.4) for
n = 0,1, 2,... .A solution {xn }
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
xn = xn+1 xn
In recent years, much research is going in the study of oscillatory behavior of solutions of third order difference equations. For more details on oscillatory behavior of difference equations, one may refer [1-22].
In this section, we present some sufficient condition for the oscillation of all the solutions of
(1.1)-(1.4). We begin with the following lemma.
Let
P(n, s, x) be defined on
N x N x R+ , N = {0,1, 2, ...}, R+ = [0, ) such that for fixed n and
s , the function P(n, s, x) is non-decreasing in x .Let {rn } be a given sequence and the sequences
{xn } and {zn } be defined on N satisfying , for all n E N ,
n 1
xn rn + P(n, s, xs ),
s =0
(2.1)
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ISSN 2229-5518
And
zn = rn +
n 1
s=0
P(n, s, zs ),
(2.2)
respectively. Then
zn s; xn
for all n E N .
This proof can be found in [18].
In addition to (H1), (H2) and (H3).assume that (H5), (H6), (H7) and (H8) hold and let
zn = xn + cn xn cr . Then, every solution of (1.1) is oscillatory.
Suppose the contrary. Then we may assume that {xn } be a non oscillatory solution of (1.1),
qn
such that xn > 0(orxn
0) for all n M
1, M > 0 is an integer and let bn = .
an
Equation (1.1) implies
bn+1
zn+1
bn zn
+ pn
xn + qn f ( xn +1 ) = 0
(2.3)
Multiplying (2.3) by
n + 1
f ( xn+1 )
and summing from M to (n
1) , we obtain
n 1 s + 1
bs+1
zs +1
n 1 s + 1
bs zs
n 1 s + 1
+ ps
xs +
n 1
(s + 1)qs = 0.
(2.4)
s = M
f ( xs+1 )
s = M
f ( xs +1 )
s= M f ( xs+1 )
s = M
But
n 1 s +1
(n +1)b x
(M +1)b x
n 1 b z
n 1 (s +1)b g (x x
) z z
bs +1 z
s +1
= n +1 n +1
M +1 M +1
s +2 s +2 +
s +2
s +2,
s +1
s +1
s +2
s =M
f (xs +1 )
f (xn +1 )
f (xMs +1M)
= sf M( xs +2 ) =
f ( xs +1 ) f ( xs +2 )
(2.5)
Also,
n 1 s + 1
(n + 1)b x
(M + 1)b x
n 1 b z
n 1 (s + 1)b g ( x x
)( z )2
bs zs
= n n
M M
s +1 s +1 +
s +1
s +2,
s +1
s +1
(2.6)
s =M f ( xs +1 )
f ( xn +1 )
f ( xM +1 )
s M=
f ( xs +2 ) s M=
f ( xs +1 ) f ( xs +2 )
Substituting (2.5) and (2.6) in (2.4), we have
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(n + 1)b z
(n +1)b z
n 1 (s +1)b g ( x
, x ) z z
(s +1)b g ( x
, x )( z )2
n +1 n +1
\ f ( xn +1 )
n n +
f ( xn+1 ) )
s = M \
s + 2 s + 2 s +1 s +1 s + 2
f ( xs +1 ) f ( xs + 2 )
s +1 s + 2 s +1 s +1
f ( xs +1 ) f ( xs + 2 ) )
n 1 b z b z
n 1 s +1 n 1
(M +1)b z
(M + 1)b z
s+2 s +2
s +1 s +1 +
p x +
(s +1)q =
M +1 M +1
M M
(2.7)
s =M \
f ( xs +2 )
f (xs+2 ) )
s =M f ( xs +1 )
s s s
s=M \
f ( xM +1 )
f ( xM +1 ) )
Using Schwarz’s inequality, we have
1
2 2
bs + 2 zs + 2 s;
n 1
(bs+ 2 )
2 n 1
zs + 2
(2.8)
s = M \
f ( xs+ 2 ) )
\ s= M
) \ s = M \ f ( xs + 2 ) ) )
1
2 2
bs +1 zs +1 s;
n 1
(bs +1 )
2 n 1
zs +1
(2.9)
s = M \
f ( xs+ 2 ) )
\ s = M
) \ s = M \ f ( xs+ 2 ) ) )
1
2 2
n 1 (s + 1)b g ( x
, x ) z z n 1
2 n 1
(s +1) g ( x
, x ) z z
s + 2 s + 2 s +1 s +1 s + 2 s;
(b )2
s + 2 s +1 s +1 s + 2
(2.10)
s = M \
f ( x
s +1
) f ( x
s + 2 )
) \ s = M
s + 2
) \ s = M \
f ( xs +1
) f ( x
s + 2 ) ) )
1
2 2
n 1 (s + 1)b g ( x
, x )( z )2 n 1
2 n 1
(s + 1) g ( x
, x )( z )4
s +1 s + 2 s +1 s +1 s;
(b )2
s + 2 s +1 s +1
(2.11)
s = M \
f ( xs +1
) f ( xs + 2 )
) \ s = M
s +1
) \ s = M \
f ( x
s +1
) f ( x
s + 2 ) ) )
And
1
2 2
(s + 1) ps xs s;
n 1
(s + 1)( p )2
2 n 1
(s + 1)
xs
(2.12)
s = M \
f ( xs +1 )
) \ s =M
) \ s =M
\ f ( xs +1 ) ) )
In view of (2.8), (2.9), (2.10),(2.11) and (2.12), the summation in (2.7) is bounded , we have
(n + 1)bn +1 zn +1
(n +1)bn zn
1
n 1 2
(bs + 2 )
1
n 1 2 2
s + 2 +
1
n 1 2
(bs +1 )
1
n 1 2 2
s +1
\ f ( xn +1 )
f ( xn+1 )
) \ s = M
) \ s = M \ f ( xs + 2 ) ) )
\ s = M
) \ s = M \ f ( xs + 2 ) ) )
1
n 1 2
n 1 (s + 1) g ( x
1
2 2
, x ) z z n 1
1
2 n 1
(s + 1) g ( x
1
2 2
, x )( z )
+ (bs +2 )
\ s =M )
\ s= M \
s +2 s +1 s+1 s + 2
f ( xs +1 ) f ( xs + 2 ) ) )
(bs +1 )
\ s= M )
\ s =M \
s +2 s +1 s +1
f ( xs +1 ) f ( xs +2 ) ) )
1
2 2
n 1
+ (s + 1)( p )2
2 n 1
(s + 1)
xs
s; (M + 1)bM +1 zM +1
(M + 1)bM zM
n 1
(s + 1)q
(2.13)
s s
\ s = M
) \ s = M
\ f ( xs +1 ) ) ) \
f ( xM +1 )
fs (MxM +1 ) ) =
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(n + 1) (b z )
In view of (H5), (H6) and (H7), we get from (2.13) that
n n � as n � .
f ( xn+1 )
Hence there exists M1
M such that (bn
zn ) 0 for n M , which implies (bn
zn )
k , k > 0
Summing the last inequality from m to (n
1) , we obtain
n 1
(bs
n 1
zs ) ( k )
s =m s =m
That is bn zn
k (n m) + bm zm
Therefore bn
zn � as n � . Hence there exists M 2
M1 such that zn
0 for n M 2
(2.14)
Rewriting (2.7), we have
(n + 1)b z
n 1 (s +1)b g (x
, x ) z z
(n +1)b z
(M +1)b z
(M + 1)b z
n +1 n +1 +
s + 2 s +2 s +1 s +1 s +2 = n n + M +1 M +1
M M
f (xn +1 )
s =M 2
f ( xs +1 ) f (xs +2 )
f ( xn +1 )
f (xM +1 )
f ( xM +1 )
n 1 n 1
(s + 1)b g ( x
, x )( z )2
M 2 1 (s + 1)b g ( x
, x ) z z
M 2 1
s + 1
(s + 1)q +
s +1 s + 2 s +1 s +1
s + 2 s + 2 s +1 s +1 s +2
p x
s s s
s = M s= M 2
f ( xs +1 ) f ( xs + 2 )
s= M
f ( xs+1 ) f ( xs +2 )
s= M f ( xs+1 )
M 2 1 (s + 1)b g ( x
, x )( z )2
M 2 1
b z b z
n 1 b z b z
n 1 s + 1
+ s +1 s +2 s +1 s +1 +
s +2 s +2
s +1 s +1 +
s +2 s +2
s +1 s +1
s s
s =M
f ( xs +1 ) f ( xs +2 )
s =M \
f ( xs +2 )
f ( xs +2 ) )
s =M 2 \
f ( xs +2 )
f ( xs +2 ) )
s =M 2 f ( xs +1 )
(2.15)
From (H1), (H7), (2.14) and (2.15), there exists an integer M 3
M 2 , such that
(n + 1)b z
n 1 (s + 1)b g ( x
, x ) z z
n+1 n +1 +
f (xn +1 )
s = M 2
s + 2 s + 2 s +1 s +1 s + 2 s;
f ( xs +1 ) f ( xs +2 )
l, l M 3
where l is a positive integer.
(n + 1)b z
n 1 (s +1)b g (x
, x ) z z
(2.16)
n +1
n +1
s + 2
s +2
s +1
s +1
s +2 l
f ( xn +1 )
s =M 2
f ( xs +1 ) f (xs +2 )
Let
un+1 =
(n + 1)
zn +1`, (2.16) becomes
u b n 1 (s + 1)b g ( x
, x ) z z
n +1 n +1
f (xn +1 )
l +
s =M 3
s +2 s + 2 s +1 s +1 s + 2 ; n M
f (xs +1 ) f ( xs +2 )
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f (x )
n 1 b f (x
)g (x
, x )(
z )u
(i.e) n +1
n +1
s +2 n +1 s + 2 s +1 s + 2 s +1
(2.17)
bn +1
s =M 3
bn +1 f (xs +1 ) f ( xs +2 )
f (x )
n 1 b f (x
) g ( x
, x )(
z )v
Also, Let
vn +1
= l n +1 +
bn +1
s =M 3
s +2 n +1 s +2 s +1 s +2 s +1
bn +1 f (xs +1 ) f ( xs +2 )
(2.18)
Using lemma 1, we have, from (2.17) and (2.18)
un +1
vn+1
(2.19)
f ( x )
n 1 b g (x
, x )(
z )v
(2.18) implies
vn +1
= n +1 l +
bn +1 \
s = M 3
s +2 s +2 s +1 s +2 s +1
f ( xs +1 ) f ( xs + 2 ) )
This implies that
lf ( x
v 3
)
; n M
(2.20)
n+1 3
n +1
From (2.19) and (2.20), we have
(n + 1)
zn+1
lf ( x )
3
bn +1
lf ( xM )
zn+1 s; + (2.21)
n b
( 1)
n+1
n 1 n 1 1
Summing (2.21) from M 3 to (n
1) , we have
zn +1 s;
lf ( x
3
)
(n + 1)b
s = M 3
s = M 3
n+1
n 1 1
That is
zn 1
zM 1 s;
lf ( xM
)
( 1)
+ 3 + 3
s = M 3
n + b 1
n +
n 1 1
zn+1 s; zM +1
lf ( xM ) (n + 1)b
s= M 3
n+1
zn = ( xn + cn xn cr ) s; 0 For sufficiently large n ,
Which is a contradiction to the fact that
xn is eventually positive. The proof is similar for the case when
xn is eventually negative. Hence the theorem is completely proved.
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ISSN 2229-5518
Examples
Example 1
Consider the difference equation
2 n x
+ nx
+ 9n
+ 18n + 5
x + xn +1 = 0
(E1)
\ n + 1
n n 3
2n2 (n +1)(n + 2)
n n(n +1)
All the conditions of Theorem 1 are satisfied. Hence every solution of equation (E1) is oscillatory.
Consider the difference equation
2 n + 1
x + nx
+ 1 n x +
( xn+1 ) = 0
(E2)
\ n + 2
n n 5 n3
n + 1 n
(n + 1)(n + 2)
All the conditions of Theorem 1 are satisfied. Hence every solution of equation (E2) is oscillatory
In addition to (H1), (H2) ,(H3)and (H4).assume that (H5), (H6), (H7) and (H8) hold and let
zn = xn + cn xn cr . Then, every solution of (1.2) is oscillatory.
In addition to (H1), (H2) and (H3).assume that (H6), (H7) and (H8) hold and let
Then, every solution of (1.3) is oscillatory.
zn = xn + cn xn cr .
In addition to (H1), (H2), (H3) and (H4).assume that (H6), (H7) and (H8) hold and let
zn = xn + cn xn cr . Then, every solution of (1.4) is oscillatory.
Proofs of Theorem 2, Theorem 3 and Theorem 4 are similar to the proof of Theorem 1 and hence the details are omitted.
Reference
[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: Discrete oscillation theory-CMIA Book Series,Volume 1,ISBN : 977-5945-19-4.
IJSER © 2011 http://www.ijser.org
International Journal of Scientific & Engineering Research, Volume 2, Issue 2, February-2011 8
ISSN 2229-5518
[3] R.P.Agarwal,Mustafa F.Aktas and A.Tiryaki: On oscillation criteria for third order nonlinear delay differential equations-Archivum Mathematicum(BANO)- Tomus 45 (2009),1-18.
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[8] B.Selvaraj and I.Mohammed ali jaffer : Oscillation Behavior of Certain Third order Linear
Difference Equations-Far East Journal of Mathematical Sciences,Volume 40, Number 2,
2010,pp 169-178.
[9] B.Selvaraj and I.Mohammed ali jaffer :Oscillatory Properties of Fourth Order Neutral Delay Difference Equations-Journal of Computer and Mathematical Sciences-An Iternational Research Journal,Vol. 1(3), 364-373(2010).
[10] B.Selvaraj and I.Mohammed ali jaffer :Oscillation Behavior of Certain Third order Non- linear Difference Equations-International Journal of Nonlinear Science(Accepted on September 6, 2010).
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[12] B.Selvaraj and I.Mohammed ali jaffer: On The Oscillation of the Solution to Third Order
Difference Equations(Journal of Computer and Mathematical Sciences-An International
Research Journal- Accepted).
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[13] B.Selvaraj and J.Daphy Louis Lovenia : Oscillation behavoir of fourth order neutral difference equations with variable coefficients- Far East Journal of Mathemati cal Sciences,Vol 35,Issue 2,2009,pp 225-231.
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