J. Comp. & Math. Sci. Vol. 1(7), 873-876 (2010)
On The Oscillation of The Solution to Third Order Nonlinear Difference Equations B. SELVARAJ and I. MOHAMMED ALI JAFFER* Dean of Science and Humanities, Nehru Institute of Engineering and Technology Coimbatore, Tamil Nadu, India. * Faculty of Mathematics, Government Arts College, Udumalpet, Tirupur, Tamil Nadu, India.642126 Email: jaffermathsgac@gmail.com ABSTRACT In this paper, we study the oscillatory behavior of the solution of the third order difference equation of the form
1 ∆ ∆ 2 yn + pn f ( yσ ( n ) ) = 0, n ∈ N = {0,1, 2,...} an Key words: Oscillation, Third order, Nonlinear, Difference equations. AMS Subject Classification: 39 A 10 n −1
INTRODUCTION
(H3)
We are concerned with the oscillatory behavior of the solution of the third order difference equation of the form
1 ∆ ∆2 yn + pn f ( yσ (n) ) = 0, n ∈ N = {0,1,2,...} an Where the following conditions are assumed to hold. (H1)
{an },{ pn } and {σ (n)} are positive
sequence and pn ≠ 0 for infinitely many values of n. (H2) σ ( n) ≤ n and lim σ ( n) = ∞. n →∞
Rn = ∑ as → ∞ as n → ∞ . s = n1
(H4) f : R → R is continuous and
xf ( x) > 0 for all x ≠ 0 and (1.1)
f ( x) ≥ L > 0. x
By a solution of equation (1.1), we mean a real sequence { yn } satisfying (1.1) for n=0, 1, 2, 3….. A solution { yn } is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called non-oscillatory. The forward difference by ∆yn
operator
= yn +1 − yn .
∆
is
Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)
defined
874
B. Selvaraj et al., J. Comp. & Math. Sci. Vol. 1(7), 873-876 (2010)
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 can refer1-9. MAIN RESULTS In this section, we present some sufficient condition for the oscillation of all the solutions of equation (1.1). Theorem 1 Assume that (H3) hold, ∆σ (n) ≥ 0 and
(∆Rσ ( s ) ) 2 LRσ ( s ) ps − = ∞, ∑ 4( s − n1 )as +1 Rσ ( s ) s = n2 for n2 ≥ n1 (1.2) ∞
Then equation (1.1) is oscillatory. Proof: Let { yn } be non-oscillatory solution of equation (1.1). Without loss of generality, we may assume that yn > 0, yσ ( n ) > o and for n ≥ n1 . From equation (1.1) we have
1 ∆ ∆ 2 yn < 0 For n ≥ n1 an 1 Since ∆ ∆ 2 yn is non-increasing ,there an exists a non negative constant k and n2 ≥ n1 such that
1 ∆ ∆ 2 yn < −k For n ≥ n2 , k > 0 an Summing the last inequality from n2 to
(n − 1) , we obtain
1 2 1 2 ∆ yn ≤ ∆ yn2 − k (n − n2 ) an an2 Letting n → ∞ , we have 1 2 ∆ yn → −∞ . an Thus, there is an integer n3 ≥ n2 such 1 2 1 2 that n ≥ n3 , ∆ yn ≤ ∆ yn3 < 0 an an3 That is ∆ 2 yn ≤ −lan , l > 0 Summing the last inequality from n3 to n −1
(n − 1) , we have ∆yn ≤ ∆yn3 − l ∑ as . s = n3
Letting n → ∞ , we have ∆yn → −∞ .Thus, there is an integer n4 ≥ n3 such that n ≥ n4 ,
∆yn ≤ ∆yn4 < 0 That is ∆yn ≤ − m , m > 0 Summing the last inequality from n4 to
(n − 1) , we have yn ≤ yn4 − m(n − n4 ) . implies that yn → −∞ as n → ∞ ,which is contradiction to the fact
This
1 2 ∆ yn > 0 an
that yn is positive . Then ∆ and
1 2 ∆ yn > 0 . an
Define wn =
∆wn =
Rσ ( n ) ∆ 2 yn an yσ ( n )
> 0, then
Rσ (n) 1 2 ∆2 yn+1 Rσ (n) ∆ ∆ yn + ∆ , yσ (n) an an+1 yσ (n)
Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)
B. Selvaraj et al., J. Comp. & Math. Sci. Vol. 1(7), 873-876 (2010)
∆wn = −
1 ∆Rσ (n) ∆ ∆2 yn + wn+1 a R σ n ( n + 1)
Rσ (n) yσ (n)
Rσ (n) ∆2 yn+1∆yσ (n)
That is ∆wn ≤ − LRσ ( n ) pn +
( ∆R )
875 2
σ (n)
4(n − n1 )an +1 Rσ ( n )
−
(n − n1 )an +1 Rσ ( n ) ∆Rσ ( n ) wn +1 − Rσ ( n +1) 2 ( n − n1 ) an +1 Rσ ( n )
an+1 yσ (n) yσ (n+1)
2
(1.3) This implies that Consider n−1
∆yn = ∆yn1 + ∑∆2 yn ≥ (n −1− n1)∆2 yn ; n ≥ n1 +1 s=n1
This implies that
∆yn +1 ≥ ( n − n1 ) ∆ 2 yn +1 ; n ≥ n2 = n1 + 1 (1.4) In view of (H2),(H4),equations (1.1) and (1.4) , we get from equation (1.3) that
∆wn ≤ −LRσ (n) pn +
∆Rσ (n) Rσ (n +1)
Rσ (n) ( ∆ yn +1 ) 2
−(n − n1 )
an +1 ( yσ (n +1) )
wn +1
−(n − n1 )
an +1 Rσ ( n ) R
2
σ ( n +1)
( wn +1 ) 2
Example (1.5)
satisfies all condition of theorem 1. Here σ (n) = n − 2 and f ( x) = x3 . Hence all solutions of equation (1.5) are oscillatory. In fact { yn } = {( −1) n } is one such solution of equation (1.5).
2
Rσ ( n +1)
Letting n → ∞ , we have, in view of (1.2) that wn → −∞ as n → ∞ , which contradicts
The difference equation ∆ ( n∆ 2 yn ) + 4(2n + 1)( yσ ( n ) )3 = 0; n > 2
2
∆Rσ ( n )
∞ (∆Rσ ( s ) ) 2 wn ≤ wn1 − ∑ LRσ ( s ) ps − 4( s − n1 )as +1 Rσ ( s ) s = n2
wn > 0 and the proof is complete.
This implies
∆wn ≤ − LRσ ( n ) pn +
2 ∆Rσ ( n ) ) ( ∆wn < − LRσ ( n ) pn − 4(n − n1 ) an +1 Rσ ( n ) Summing the last inequality from n2 to (n − 1) , we have
wn +1
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Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)
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B. Selvaraj et al., J. Comp. & Math. Sci. Vol. 1(7), 873-876 (2010)
oscillation theory-CMIA Book Series, Volume 1, ISBN : 977-5945-19-4. 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, pp 169-178 (2010). B.Selvaraj and I.Mohammed Ali Jaffer: Oscillatory Properties of Fourth Order Neutral Delay Difference EquationsJournal of Computer and Mathematical Sciences-An Iternational Research Journal,Vol. 1(3), 364-373 (2010). 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). B.Selvaraj and I.Mohammed Ali Jaffer :
Oscillation Theorems of Solutions For Certain Third Order Functional Difference Equations With DelayBulletin of Pure and Applied Sciences (Accepted on October 20, (2010). 7. E.Thandapani and B.Selvaraj: Existence and Asymptotic Behavior of Non oscillatory Solutions of Certain Nonlinear Difference equation- Far East Journal of Mathematical Sciences 14(1), pp: 9-25 (2004). 8. E.Thandapani and B.Selvaraj: Oscillatory and Non-oscillatory Behavior of Fourth order Quasi-linear Difference equation -Far East Journal of Mathematical Sciences 17(3), 287307 (2004). 9. E.Thandapani and B. Selvaraj: Oscillation of Fourth order Quasi-linear Difference equation-Fasci culi Mathematici Nr, 37, 109-119 (2007).
Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)