J. Comp. & Math. Sci. Vol.2 (3), 567-571 (2011)
Oscillatory Properties of Certain First and Second Order Difference Equations B. SELVARAJ1 and J. DAPHY LOUIS LOVENIA2 1
Dean of Science and Humanities, Nehru Institute of Engineering and Technology, Coimbatore, Tamil Nadu, India. 2 Department of Mathematics, Karunya University, Karunya Nagar, Coimbatore, Tamil Nadu, India. ABSTRACT In this paper some sufficient conditions for the oscillation of all solutions of certaindifference equations are obtained. Examples are given to illustrate the results.
(H6) p n < a n
1. INTRODUCTION We consider the oscillatory properties of the solution of the linear difference equations of the form
∆(a n x n ) + q n xn = 0
(1.1)
∆(a n x n ) + qn x n+1 = 0
(1.2)
∆ ( a n x n ) + p n ∆x n + q n x n = 0
(1.3)
2
∆2 ( a n x n ) + p n ∆x n + q n x n +1 = 0 The following assumed to hold. (H1)
{an } , { pn }
conditions
(H3) a n+1 + q n < 0 (H4) p n ≥ 2a n+1 (H5) a n + q n > p n
are
By a solution of equations (1.1)(1.4), we mean a real sequence {xn } satisfying (1.1)-(1.4) for n=0,1,2,3,…. A solution {xn } is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. ∆ is the forward difference operator defined by ∆x n = xn+1 − xn . 2. MAIN RESULTS
{qn }
and sequences for all values of n. (H2) a n ≤ q n
(1.4)
(H7) p n + q n > 2a n +1
are
real
Theorem 1 In addition to (H1), assume that (H2) holds. Then every solution of (1.1) is oscillatory, and one solution of equation (1.1) is n −1 a − qs xn = x0 ∏ s s = 0 a s +1
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
(2.1)
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B. Selvaraj, et al., J. Comp. & Math. Sci. Vol.2 (3), 567-571 (2011)
oscillatory. A solution of equation (E1) is
Proof: From equation (1.1) we get,
(−1) n xn = , x0 = 1 n!
a n+1 xn+1 − an x n + q n xn = 0 a − qn xn xn +1 = n an +1
(2.2)
Multiplying equation (2.2) by x n , we have
xn x n+1
a − qn = n a n+1
2 xn
(2.3)
From (H1),(H2), and equation (2.3) we have
Theorem 2 In addition to (H1), assume that (H3) holds. Then every solution of (1.2) is oscillatory, and one solution of equation (1.2) is n −1 as xn = x0 ∏ s = 0 a s +1 + q s
(2.4)
Proof: From equation (1.2) we get
xn xn+1 ≤ 0
a n+1 xn+1 − a n xn + qn x n+1 = 0
Hence any solution of equation (1.1) is oscillatory.
an xn xn+1 = a + q n n+1
(2.5)
Multiplying equation (2.5) by x n , we have Again from equation (2.2),we have
x n +1 a n − q n = xn a n +1
Taking the product from 0 to (n-1) on both sides, we obtain
x n a0 − q0 a1 − q1 a n−1 − q n−1 ... = x0 a1 a2 an n −1 a − qs ⇒ x n = x0 ∏ s s = 0 a s +1 Example 1: Consider the difference equation
∆(nxn ) + (n + 1) xn = 0
(E1) From equation (1.1) and equation (E1), an = n; qn = (n + 1) which is condition (H2), hence any solution of (E1) is
an 2 xn xn xn+1 = an +1 + q n
(2.6)
From (H1),(H3) and equation (2.6), we have xn xn+1 ≤ 0 . Hence any solution of equation (1.2) is oscillatory. Again from equation (2.5), we have
xn +1 an = xn a n +1 + q n Taking the product from 0 to (n-1) on both sides, we obtain
xn a0 a1 an−1 ... = x0 a1 + q0 a2 + q1 an + qn−1 n −1 as ⇒ xn = x0 ∏ s = 0 a s +1 + q s
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
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B. Selvaraj, et al., J. Comp. & Math. Sci. Vol.2 (3), 567-571 (2011)
Example 2 Consider the difference equation
∆((n + 1) xn ) − (n + 3) xn+1 = 0
(E2)
from equation (1.2) and equation (E2),
,
which is condition (H3). Hence any solution of (E2) is oscillatory. A solution of equation (E2) is
xn =
n! ; x0 = 1 . (−1) n
Example 3 Consider the difference equation
+ (n 3 + 6n 2 + 11n + 6) xn = 0
(2.7)
p − 2a n +1 xn +1 ⇒ xn + 2 = − n an+2 a − pn + qn xn − n an+ 2
(E3)
From equation (E3) and equation (1.3) an = n; pn = ( 2 n + 3); q n = ( n 3 + 6 n 2 + 11n + 6); p n = ( 2 n + 3) > 2 a n +1 = 2 n + 2; an + qn = n 3 + 6 n 2 + 12 n + 6 > p n = 2 n + 3;
Multiplying the above equation by x n +1
, we
Conditions (H6) and (H7) are satisfied. One such solution of (E3) is
have p − 2an +1 2 x n +1 xn +1 xn + 2 = − n an+2 a − p n + qn xn xn +1 − n an + 2
(2.10)
∆2 (nxn ) + (2n + 3)∆xn
Proof: From equation (1.3) + (an − pn + q n ) xn = 0
p − 2an +1 2 x n +1 xn +1 xn + 2 ≤ − n an + 2 a − p n + qn 2 x n +1 < 0 − n an + 2
Hence xn +1 xn + 2 ≤ 0 , this completes the proof of the theorem.
Theorem 3 In addition to (H1), assume that (H4) and(H5) hold. Then every solution of (1.3) is oscillatory.
a n +1 xn + 2 + ( pn − 2an +1 ) x n +1
(2.9)
and for xn ≥ x n +1
a n = n + 1; q n = −(n + 3); an +1 = n + 2; a n +1 + qn = −1 < 0
p − 2an +1 2 x n +1 xn +1 xn + 2 ≤ − n an + 2 an − p n + qn 2 x n < 0 − an + 2
xn =
(2.8)
From condition (H1),(H4) and (H5), we get for xn ≤ x n +1
(−1)n , xo = 1 n!
Theorem 4 In addition to (H1), assume that (H6) and (H7) hold. Then every solution of (1.4) is oscillatory. Proof: From equation (1.4)
∆2 ( a n x n ) + p n ∆x n + q n x n +1 = 0
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B. Selvaraj, et al., J. Comp. & Math. Sci. Vol.2 (3), 567-571 (2011)
∆ ( an +1 xn +1 − a n xn ) + pn ( x n +1 − xn ) + qn x n +1 = 0 a n + 2 x n + 2 − 2 a n +1 xn +1 + an xn + p n xn +1 − pn xn + q n xn +1 = 0 p + q n − 2 a n +1 a − pn x n +1 − n xn + 2 = − n a a n+ 2 n+2
xn
p n + q n = − n + n 2 + 9 n + 11 −
Multiplying the above equation by x n +1 , we have p + qn − 2an +1 2 x n +1 xn +1 xn + 2 = − n an + 2 an − pn xn xn +1 − an + 2
Hence proof.
Conditions (H6) and (H7) are satisfied. and a solution of (E3) is
(2.11)
xn =
n! ; x0 = 1 (−1) n
Example 5 Consider the difference equation
(2.12) ∆2((n +1)xn ) − n∆xn + (2n2 + 6n + 6 +
and for xn ≥ x n +1 p + qn − 2an +1 2 x n +1 xn +1 xn + 2 ≤ − n an + 2 an − pn 2 x n +1 < 0 − an + 2
n2 > 2 a n + 1 = 2 ( n + 2 ); n +1
Hence any solution of (E3) is oscillatory,
From (H1),(H2) and (H3),(H8) and (H9), we get for xn ≤ x n +1 p + qn − 2an +1 2 x n +1 xn +1 xn + 2 ≤ − n an + 2 an − pn 2 x n < 0 − an + 2
n2 a n = n + 1; p n − n ; q n = n 2 + 9 n + 11 − ; n +1 p n = − n < a n = n + 1;
xn+1 = 0, n ≥1
(2.13)
xn +1 xn+ 2 ≤ 0 , this completes the
1 ) n+2
(E5)
From this equation and equation (1.4)
an = n + 1; pn − n; qn = (2n 2 + 6n + 6 + pn = −n < an = n + 1;
1 ); n+2
pn + qn > 2an +1 = 2(n + 2);
Example 4 Consider the difference equation ∆2 ((n + 1) xn ) − n∆ (xn ) + 2 2 n + 9n + 11 − n xn +1 = 0 n + 1
(E4)
From equation (1.4) and equation (E3),
Conditions (H6) and (H7) are satisfied. Hence any solution of (E4) is oscillatory. One such solution of (E4) is
(−1) n xn = ; x0 = 1 n!
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
B. Selvaraj, et al., J. Comp. & Math. Sci. Vol.2 (3), 567-571 (2011)
REFERENCES 1. R. P. Agarwal : Difference Equation and Inequalities, Marcel Dekker; Second Edition, New York, (2000). 2. R. P. Agarwal, M. Bohner, S.R. Grace and D. O. Regan, Discrete Oscillation Theory, Hindawi Publishing Corporation, (2005). 3. Sh.Salem, K. R. Rasalam : Oscillation of some second Order Damped Difference Equations-IJNS.Vol.5, No.3, 246-254 (2008). 4. E. Thandapani and B. S. Lalli: Oscillations Criteria for a Second Order Damped Difference Equations, Appl. Math. Lett.,Vol.8,N0.1,1-6 (1995). 5. E.Thandapani, I. Gyori, and B. S. Lalli: An Application of Discrete Inequality to Second Order Nonlinear Oscillation. J.Math.Anal Appl.186, 200-208 (1994). 6. E.Thandapani and S. Pandian : On The Oscillatory Behavior of Solutions of Second Order Non â&#x20AC;&#x201C;linear Difference Equations, ZZA 13, 347-358 (1994). 7. E. Thandapani and S. Pandian: Oscillation Theorem for Non-linear Second Order Difference Equations with a Nonlinear Damping Term, Tamkang J. Math.,26, 49-58 (1995).
571
8. E. Thandapani: Asymptotic and Oscillatory Behavior of Solutions of Nonlinear Second Order Difference Equations., Indian J. Pure and Appl. Math 24, 365-372 (1993). 9. E.Thandapani and K.Ravi : Oscillation of Second Order Half linear Difference Equations, Appl. Math. Lett. 13, 43-49 (2000). 10. E.Thandapani and B. Selvaraj: Oscillatory Behavior of Solutions of Three dimensional Delay Difference System, Radovi Mathematicki, Vol.13, 39-52 (2004). 11. E.Thandapani and B. Selvaraj: Existence and Asymptotic Behavior of Non oscillatory Solutions of Certain Nonlinear Difference Equation â&#x20AC;&#x201C; Far East Journal of Mathematical Sciences 14(1), 9-25 (2004). 12. 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). 13. E.Thandapani and B. Selvaraj: Oscillation of Fourth Order Quasi-linear Difference Equations, Fasci culi Mathematici 37,109-119 (2007).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)