Cmjv01i04p0443 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol. 1(4), 443-447 (2010).

Oscillation behavior of certain fourth order neutral difference equations B. SELVARAJ and J. DAPHY LOUIS LOVENIA Department of mathematics Karunya University, Karunya Nagar, Coimbatore-641 114, Tamil Nadu, India. e-mail: selvaraj@karunya.edu, daphy@karunya.edu ABSTRACT In this paper, we establish the sufficient conditions for oscillation of fourth order neutral difference equation of the form

2 ( rn 2 ( y n  p n y n  k ))  q n y n l 3  0 by using comparison method. Example is provided to illustrate the results. Keywords : Oscillation, Neutral difference equations, comparison method. AMS Classification :39 A 11

1. INTRODUCTION

real sequences with {q n } not identically

Consider the following fourth order non linear neutral difference equation of the form

2 ( rn 2 ( y n  p n y n  k ))  q n y n l 3  0 (1.1)

zero for infinitely many values of n. (b) There is a positive constant p such that 0  p n  p  1 and k and l are positive integers.

where n  N ( n0 )  {n0 , n0  1, n0  2,...},

,...},n0 is a non-negative integer and  is the forward difference operator defined by y n  y n 1  y n. . The following conditions are assumed to hold: (a)

{ pn }

and {q n } are non-negative e

that

r

 .

n  n0 n

Let   max{ k , l} . By a solution

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B. Selvaraj et al., J.Comp.&Math.Sci. Vol.1(4), 443-447 (2010)

of (1.1) we mean a real sequence

{ yn }

which is defined for all n  N 0   and satisfies equation (1.1) for all n  N

N (n0 ) . As it is customary, a solution

{ y n } of (1.1) is said to be oscillatory if the terms yn of the sequence are not eventually positive or not eventually negative; otherwise it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

solution of the equation (1.1), then there are only the following two cases for large n,

( I ) y n  0, z n  0, z n  0, rn 2 z n  0,

 (rn 2 z n )  0. ( II ) y n  0, z n  0, z n  0, rn 2 z n  0,  (rn 2 z n )  0 The proof follows from Discrete Kneser’s Theorem [1,Theorem 1.7.11] Lemma 2.2 Let {yn} be a positive solution of the equation (1.1) and not identically zero. Then for every

In recent years, there has been an increasing interest in the study of oscillatory and asymptotic behavior of  ,0    1 , there exists a large positive e solutions of fourth order difference equations2-20 and references cited therein. integer N such that for all n  N  , Following this trend, in this paper we  yn  n(3)  (rn  2 yn ), where n ( 3)  n obtain some sufficient conditions for the 3! oscillation of all the solutions of equation (1.1) using comparison method. An n( n  1)( n  2) . example is provided to illustrate the Proof. The proof can be found result. in [16] and [20]. 2. Main results Lemma2.3 If {yn}

Let {xn} be a real sequence. We define a sequence {zn} by z n  y n  p n y nk , (2.1)

is an eventually positive solution of (1.1) , then there exists an integer n  N (n0 ) such that (1  p n ) z n  y n  z n for n  N .

n  N (n0 ) where {p n} and k are as defined as earlier. In this section we present some sufficient conditions for the oscillation of all solutions of equation (1.1). Lemma2.1 Let {yn} be a positive

Proof. From definition of we have

zn  yn

zn ,

for n  N . Hence

y n  z n  p n y n k y n  z n  pn znk y n  (1  p n ) z n

for n  N .

Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)

B. Selvaraj et al., J.Comp.&Math.Sci. Vol.1(4), 443-447 (2010) Theorem 3.1 Assume that there exists a constant 0  0  1 , such that the first order delay difference

zn 

 ( 3) n  ( rn 2 z n ), 3!

445

(3.3)

From the equation (2.1)

0 y n l  3  z n l 3  p n l  3 y n l  3 q n ( n  l  3) (3) equation y n  3! and we have 2 2 [1  pn l 3 ] y n l 3  0 (3 .1)  ( rn  z n )  q n [ z n l  3  p n  l  3 y n l  3 ]  0 for all large n. is oscillatory. Then equation (1.1) is oscillatory. Proof: Let {yn} be an eventually positive solution of (1.1) and define

z n  y n  p n y n k

Since

z n  y n and z n  0 , we obtain

2 (rn 2 z n )  q n [1  p n l  3 ] z n l  3  0 for all sufficiently large n. Using (3.3),

(3 .2)

Then from hypotheses (a) and (b) there exists an integer n1  n0 such that

for every 0    1,

2 (rn 2 z n ) 

[1  p n l  3 ][rn l 3 ( 2 z nl  3 )]  0.

y n  0 and 2 ( rn 2 z n )  0 for all n  n1 . By Lemma 2.1, there exists for some large n2  n1 , such that either

for all large n. Let v n   ( rn 2 z n ) . Thus {v n } satisfies

y n  0, z n  0, z n  0, rn 2 z n  0,

,  ( rn 2 z n )  0. OR y n  0, z n  0, z n  0, rn 2 z n  0,  (rn 2 z n )  0 Hence lim n z n  0. Therefore by Lemma 2.2 for every  ,0    1 , there e exists a

N

   q n (n  l  3) (3) 3! 3!

such that for all n  N  ,

 q n (n  l  3) ( 3) 3!  0 , for n large

v n  [1  p n l 3 ]v n l 3 enough and for every 0    1.

Using result from [1], the difference equation

 q n (n  l  3) ( 3) [1  p n l 3 ]w 3! wn l 3  0.

wn 

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B. Selvaraj et al., J.Comp.&Math.Sci. Vol.1(4), 443-447 (2010)

has an eventually positive solution for every 0    1. This contradicts the fact that (3.1) is oscillatory. When {yn} is eventually negative solution, {-yn} will be an eventually positive solution. Proof of Theorem 3.1 is complete.

81-88 (200). 3. R.P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York (2000). 4. R.P. Agarwal, E. Thandapani and P.J.Y. Wong, Oscillation of higher order neutral difference equations, Appl. Math. Lett., 10(1), 71-78 3. Example: Consider the difference (1997). 5. S.S. Chang, on 9a class2 of fourth 1 equation  2 [ n(n  1) 2 ( yn  yn 1 )]  8(2order n 2  6linear n  5) reccurence  n (9n equations,  21n  14) yn 1  0, n  1 n 2 2 Internat. J. Math & Math. Sci. 7, 1 9   131-149 (1984).  1) 2 ( yn  n yn 1 )]   8(2n 2  6n  5)  n 4 (9n 2  21n  14)6.  0, n  1and E. Schmeidal, On 1 Popenda  ynJ. 2 2   the solution of fourth order difference equations, Rocky Mountain.  (4 .1)  6n  5)  (9n  21n  14)  yn 1  0, n  1 J. Math. 25, 1485-1499 (1995).  7. B. Selvaraj, M. Mallika Arjunan and V. Kavitha–‘Existence of Solutions 1 2 Here p n  n , q n  8(2n  6n  5)  for Impulsive Nonlinear Differential 2 Equations with Nonlocal Conditions’ –J KSIAM, Vol. 13, No. 3, September 9 2 ( 9 n  21 n  14 ), r  n ( n  1 ), pp. 203-215 (2009). n 2 n4 8. B. Selvaraj and J. Daphy Louis Lovenia – ‘Oscillation Behavior of k  1, l  2. It is easy to see that all Fourth Order Neutral Difference solutions of equation (4.1) are osciEquations with Variable Coefficients’ llatory. –Far East Journal of Mathematical Sciences (FJMS), Vol. 35, Issue n In fact { y n }  { 1 } is one such 2, pp. 225-231 (2009). solution of equation (4.1) 9. B. Selvaraj and I. Mohammed Ali Jaffer – ‘Oscillation Behavior of REFERENCES Certain Third Order Linear Difference Equations’, accepted for publication 1. R.P. Agarwal, M. Bohner, S.R. Grace in “Far East Journal of Mathemaand D.O. Regan, Discrete Oscillation tical Sciences” in February 5, (2010). Theory, Hindawi Publishing Corpo- 10. B. Selvaraj and G. Gomathi Jawahar ration (2005). – ‘Asymptotic Behavior of Fourth 2. R.P. Agarwal and S.R. Grace, OsciOrder Non-linear Delay Difference llation of higher order difference Equation’, accepted for publication equations, Appl. Math. Lett., 13, in “Far East Journal of MathemaJournal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)

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Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)