J. Comp. & Math. Sci. Vol. 1(5), 566-571 (2010)
Non Oscillation of Second Order Neutral Delay Difference Equations B. SELVARAJ and G.GOMATHI JAWAHAR Department of Mathematics Karunya University, Karunya Nagar, Coimbatore – 641 114 Tamil Nadu, India. Email : Selvaraj@karunya.edu. Email:jawahargomathi@yahoo.com
ABSTRACTS In this paper we deal with the non oscillatory behavior of all solutions of the second order neutral delay difference equations . Example is provided to illustrate the main results. Key words: Non oscillation, Difference equations , Neutral Delay. Subject classification : 2000 mathematics 39 A11.
INTRODUCTION Consider the neutral delay difference equation of the form,
(1:1) Where k,l are non negative integers, qn (n= 0,1,2….) are real numbers, and denotes the forward difference operator defined by yn = yn+1-yn In addition to the above, we assume the following.
H1 : qn >0, eventually positive . H2 : yn-l > yn-l+k By a solution of (1.1) we mean a real sequence {yn} which satisfies the equation (1.1) for all nN(no). A solution {yn} 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 non oscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory. In the past few years, there has
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
B. Selvaraj et al., J. Comp. & Math. Sci. Vol. 1(5), 566-571 (2010)
been an increasing interest in the study of oscillatory and non oscillatory behavior of difference equations. For example see [2,3,4,5,6,7,8,9,10,11]. Following this trend, in this paper we obtain some sufficient conditions for the nonoscillation of all solutions of equation (1.1). An example is provided to illustrate the main results. MAIN RESULTS In this section, we present some sufficient condition for the non oscillation of all solutions of equation (1.1) Theorem 2:1 Assume that
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Case I zn<0 and zn>0 , for n>n1 from (1.1), 2 zn = qn yn-l (zn+1 - zn) = qn yn-l zn = zn+1 - qn yn-l ==> zn = zn+2 - zn+1 - qn yn-l Hence zn - zn+k – qn+k yn- l+k + qn yn-l – yn+2-k + yn+1-k +yn+2+k – yn+k+1 = zn+2 – zn+1 – qn yn-l – (zn+k+2-zn+k+1-qn+k yn+k -l) – qn+k yn-l+k +qn yn-l – yn+2-k + yn+1-k +yn+2+k – yn+k+1 =0 (1.3) Suppose that lim sup qn+k = < , for some n > n2. Then from (1.3), takin n we have, zn - zn +k + (yn –l – yn-l+k) – yn+2 –k+ yn+1-k + yn+2+k-yn+k+1, < 0 for ne > n2. (1.4)
=
and K {1,2….}, 1 {0,1,2….} Then every non oscillatory solution of equation (1.1) tends to zero as n tends to . Proof Let {yn} be a nonoscillatory solution of equation (1.1). Let us assume that {yn} is eventually positive. Thus, there exists an integer n1>1 such that qn>0 and yn-k > 0 for n > n1. Let zn = yn+yn-k (1.2) 2 Then from (1.1), zn = qn yn-l 2 zn > 0 Hence there are 2 possibilities zn < 0 , zn > 0
Now we want to prove that lim z n= o n Suppose lim zn = a > 0, then n obviously lim yn= a>o. n Summing both sides of (1.4) from n2 to N, we have zN+1 – zn2 – (zN+k+1 – zn2 +k) N < - a as N n2 Which implies that zn contradiction.
as n
,
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a
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B. Selvaraj et al., J. Comp. & Math. Sci. Vol. 1(5), 566-571 (2010)
Hence lim zn = 0 n
Hence lim Zn = 0 n
Which implies that lim yn = 0 n
Which implies that
Hence every non oscillatory solution of equation (1.1) tends to Zero as n .
Hence every non oscillatory solution of (1.1) tends to Zero as n .
Case II zn >o and zn > o for n > n3.
Theorem 2.2 Assume that qn = . n= k
Suppose that lim inf qn = 1< for some n n > n 4. Substituting in (1.3), we have zn - zn +k + 1 (yn –l – yn-l+k) – yn+2 –k - yn+2-k + yn+1-k+yn+2+k- - yn+k+1 > 0 for n > n4 (1.5) Now we want to prove that lim zn= o n
Let q*n = min (qn, qn+k) when zn < 0 and q*n = max (qn, qn+k) When zn > 0.
lim yn = 0 n
Suppose
lim zn = a1 > 0, then n obviously lim yn = a1 > 0. n Summing both sides of (1.5) from n4 to N, we have ZN+1 - Zn4 - (ZN+k+l - Zn4) N > -a1 as N n4 Which implies that Zn contradiction.
as n
,
a
Then every non oscillatory solution of equation (1.1) tends to zero as n tends to . Proof Let {yn} be a eventually positive solution of (1.1). Let zn = yn+yn-k from (1.1), 2 zn = qn yn - l Hence 2 zn > 0 Hence there are 2 possibilities. zn < 0 , zn >0 Case I Suppose zn < 0. Hence zn is eventually positive and decreasing So lim zn = M > 0 exists and is finite. n 2 since zn - qn yn-l =0, we have
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
B. Selvaraj et al., J. Comp. & Math. Sci. Vol. 1(5), 566-571 (2010)
2 zn+ k - qn+k yn+k-l = 0
(1.6)
Subtracting ( 1.6) from (1.1) , 2 zn+ k - qn+k yn+k-l - (2 zn - qn yn-l ) =0 2 zn+k - 2 zn - qn+k yn+ k-l + qn yn-l =0 zn+k+2 -2 zn+k+1 + zn+k - zn+2 + 2 zn+1 - zn - qn+k yn+k-l + qn yn-l = 0 (1.7) Since q*n = min ( qn, qn+k), we have zn+k+2 - 2 zn+k+1 + zn+k - zn+2 + 2 zn+1 - zn + qn* (yn-l - y n+k-l ) < 0, For some n > n5. Hence zn+k+2 - 2 zn+k+1 + zn+k - zn+2 + 2 zn+1 - zn < - qn* ( y n-l - yn+k-l) (1.8) Summing both sides from n5 to N, we have N ( zN+k+2 - 2 z N+k+1 + z n+k - z n+2 n5 N +2 zn+1 - zn) < - qn ( yn-l - y n+k-l) as N . n5
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Hence z n is eventually positive and increasing sequence. So, lim zn = M > 0 exists and is finite. n Here q*n = max (qn, qn+k) Hence equation (1.7) becomes, zn+k+2 - 2 zn+k+1 + zn+k -zn+2 + 2zn+1 - zn > - qn (yn-l - yn+k-l) For some n > n6. Taking summation from n6 to N, we have N (zn+k+2 - 2 zn+ k+1 + zn+k n6 - zn+2 + 2 zn+1 - zn) > - qn*(yn-l - yn+ k-l)
as N
Hence lim zn n
as n .
Which is a contradiction, since Which implies that zn as n which is a contradiction. Hence lim zn= 0 n Therefore
lim yn=0. n
Case II Suppose zn > 0
,
lim zn = M > 0 and is finite. n Therefore
lim Zn = 0 n
Hence lim yn=0 n Therefore every non oscillatory solution of (1.1) tends to zero as n .
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
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EXAMPLES
2.
R.P Agarwal, M.M.S Manuel & E. Thandapani, 'oscillation and non
1. Consider the difference equation
2 (yn+yn-2) - 5 yn-3 = 0 for n > 3
oscillation of second order neutral Delay Difference equations',Appl.
(3.1)
Maths let. Vol 10 , pp 103-109.
32 3.
Basak Karpuz, 'some oscillation and non oscillation criteria for neutral
Here k= 2:, l=3:, qn = 5 /32
delay difference equations with It is easy to see that all conditions of theorem (2.1), (2.2) are satisfied. Hence all non oscillatory solutions of (3.1) goes to zero as n . One such solution is 1/2n.
positive and negative coefficients', Computers and Mathematics with Applications, vol. 57, Issue 4, pages 633- 642. 4.
R.S Dahiya, 'Non oscillation generating delay terms in even order
2. Consider the difference equation
differential equations', Hiroshima
(yn+yn-2) - 4n+8 2
n -5n+4
yn = 0 for n>4 (3.2)
Maths J, Vol 5, Pages ( 385-394) 5.I
of delay differential equations with
4n+8 Here k=2; l=0; qn = n2 -5n+4 It is easy to see that all conditions of theorem (2.1), (2.2) are satisfied. Hence all non oscillatory solutions of (3.2) goes to zero as n . one such solution is 1 n REFERENCES 1.
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B. Smith and W.E. Taylor, Jr. 'oscillation and non oncillation theorem for some mixed difference equations', Internet
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Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
B. Selvaraj et al., J. Comp. & Math. Sci. Vol. 1(5), 566-571 (2010)
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Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)