J. Comp. & Math. Sci. Vol.4 (5), 350-355 (2013)
Oscillatory Solutions of Certain Fourth Order Nonlinear Difference Equations B. SELVARAJ1, P. MOHANKUMAR2 and A. BALASUBRAMANIAN3 1
Dean of Science and Humanities, Nehru Institute of Engineering and Technology, Coimbatore, Tamil Nadu, INDIA. 2 Professor of Mathematics, Department of Science and Humanities, Arupadai Veedu Institute of Technology, Paiyanoor, Tamil Nadu, INDIA. 3 Assistant Professor in Mathematics, Arupadai Veedu Institute of Technology, Paiyanoor, Tamil Nadu, INDIA. (Received on: September 19, 2013) ABSTRACT The objective of this paper is to study the oscillatory behavior of third order nonlinear neutral delay difference equation of the form
1 ∆2 ∆2 ( y n + p n y n − k ) + q n f ( yσ ( n ) ) = 0 . Example is an given to illustrate the results. Keywords: difference equations, oscillation, nonlinear. AMS Subject Classification: 39A11.
1. INTRODUCTION We are concerned with the oscillatory properties of all solutions of a third order nonlinear neutral delay difference equation of the form 1 ∆2 ∆2 ( y n + p n y n− k ) + q n f (yσ ( n ) ) = 0 ,(1.1) an
where ∆ is the forward difference operator defined by ∆y n = y n +1 − y n , k is a fixed nonnegative integer and {a n }, {p n }, {q n } are real sequences with respect to the difference equation (1.1), throughout. We shall assume that the following conditions hold: (H1) {a n }, {p n }, {q n }are real sequences and
a n ≥ 0 for infinitely many values of n.
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B. Selvaraj, et al., J. Comp. & Math. Sci. Vol.4 (5), 350-355 (2013)
(H2) f : R → R is
continuous
yf ( y ) > 0, for all y ≠ 0 .
and
(H3) σ (n ) ≥ 0 is an integer such that
limσ (n) = ∞ .
(H4) Rn =
∑a
s = n0
s
→ ∞ as n → ∞ .
By a solution of equation (1.1), we mean a real sequence {y n }satisfying (1.1)
for n ≥ n0 . A solution {y n } is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is called nonoscillatory. For more details on oscillatory behavior of difference equations, one can refer1-27. 2. MAIN RESULTS Theorem 1: In addition to the conditions (H1), (H2), (H3), (H4), if the conditions (C1) p n ≥ 0 and
∞
∑p
s = n0
s
= ∞ , (C2)
∞
∑ q s = ∞ ,(C3) lim inf f (u ) ≥ 0 .
s = n0
y n > 0, ∆y n > 0, y n − m ≥ 0, ∆y n − m ≥ 0 , for all n ≥ n1 . Set z n = y n + p n y n − k , then in view of (C1),
n→∞
n −1
generality we can assume that there exists an integer n1 ≥ n0 such that
u →∞
Then every solution of equation (1.1) is oscillatory.
z n > 0, ∆z n ≥ 0 , for all n ≥ n1 . From equation (1.1), we have
1 ∆2 ∆2 z n = −q n f ( yσ ( n ) ) , an for all n ≥ n1 .
(2.1)
In view of the conditions (H2), (H3), (C2) and from the equation (2.1), we obtain
1 ∆ ∆2 z n is eventually non-increasing. an 1 2 ∆ z n ≥ 0 , for We first show that ∆ a n n ≥ n1 . Suppose that, there exists an integer n2 ≥ n1 and k1 > 0 such that 1 ∆ ∆2 z n ≤ − k1 , for all n ≥ n2 . an
(2.2)
Summing the inequality (2.2) from n2 to n − 1 , we have
Proof: Suppose that the equation (1.1) has non-oscillatory solution {y n }is eventually positive.Then there is a positive integer n 0 such that yσ ( n ) ≥ 0 , for n ≥ n0 implies that
{y n }
is nonoscillatory. Without loss of
1 2 1 2 ∆ zn − ∆ z n2 ≤ −k1 (n − n2 ) , an a n2 for all n ≥ n2 . Therefore,
1 2 ∆ z n → −∞ as n → ∞ . an
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(2.3)
B. Selvaraj, et al., J. Comp. & Math. Sci. Vol.4 (5), 350-355 (2013)
Summing the inequality (2.6) from n4 to n − 1 , we have
Then there exists an integer n3 ≥ n 2 and k 2 > 0 such that
1 2 ∆ z n ≤ − k 2 , for all n ≥ n3 . an
(2.4)
1 2 1 n −1 1 ∆ ∆2 z n − ∆ ∆ z n4 + ∑ q s f (L ) ≤ 0 , an 2 s =n an 4 4
for all n ≥ n4 .
Summing the inequality (2.4) from n3 to n − 1 , we have n −1
∆z n ≤ − k 2 ∑ a s , for all n ≥ n3 .
352
(2.5)
s = n3
(2.7)
In view of (C2) and (C3), from inequality (2.7), we find that ∞ ≤ 0 as n → ∞ , which is a contradiction. Case(ii): L = ∞ .
In view of the condition (H4), and from the inequality (2.4), we obtain ∆z n → −∞ as n → ∞ , which is a
integer
In view of (C3), there exists an n 4 ≥ n3 and k 3 > 0 such that
f ( yσ ( n ) ) > k 3 , for all n ≥ n5 .
contradiction to the fact that ∆z n ≥ 0 , for n . This shows that all large
Therefore, from equation (1.1), we obtain
1 ∆ ∆2 z n ≥ 0 , for all large n . an Let L = lim y n . Then L is finite or infinite.
1 ∆2 ∆2 z n + q n k 3 ≤ 0 , for all n ≥ n5 . an (2.8)
n →∞
Case(i): L > 0 is finite. In view of (H2), (H3), we have
lim f ( y ) = f (L ) > 0 . n →∞
σ (n)
This implies that
f ( yσ ( n ) ) >
1 f (L ) > 0 , for all large n . 2 Then there exists an integer n 4 ≥ n3 and from equation (1.1), we obtain
1 1 ∆2 ∆2 z n + q n f (L ) ≤ 0 , an 2 for all n ≥ n4 .
(2.6)
The remaining proof is similar to that of case(i), and hence we omitted. Thus in both cases we obtained that {y n } is oscillatory. In fact y n < 0, y n − m < 0 , for all large n , the proof is similar, and hence we omitted. This completes the proof. Corollary 1: In addition to the conditions (H1), (H2), (H3), (H4), if the conditions of Theorem1 hold. Then every bounded solution of equation (1.1) is oscillatory. Proof: Proceeding as in the proof of Theorem 1 with assumption that is {y n }
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B. Selvaraj, et al., J. Comp. & Math. Sci. Vol.4 (5), 350-355 (2013)
bounded non-oscillatory solution of equation (1.1). Therefore, from inequality (2.6) of Theorem 1, we find that
1 1 Rn ∆2 ∆2 z n + Rn q n f (L ) ≤ 0 , for an 2 (2.9) all n ≥ n4 . By the definition of Rn and from the inequality (2.9), we find that
1 ∆ ∆2 z n ≤ − Rn q n f (L ) , for all n ≥ n4 . an (2.10) In view of (C2), (C3) and (H4), we have
1 ∆ ∆2 z n → −∞ an
as
n → ∞,
which is a contradiction to the fact that
1 ∆ ∆2 z n ≥ 0 , for all large n . This an shows that sequence {y n } is a bounded oscillatory solution of equation (1.1). This completes the proof. 3. EXAMPLE Example1: Consider the difference equation 1 ∆2 2n∆2 y n + y n −1 + 8(n + 1) y n + 2 = 0, n > 0 2
(3.1) 1 Here an = 2n, pn = , qn = 8(n + 1), . 2 and f ( yσ ( n ) ) = yn+ 2 , k = 1.
All the conditions of the Theorem1 are satisfied, and hence all solutions of equation (3.1) are bounded oscillatory. One such solution of equation (3.1) n +1 . is {y n } = (− 1)
{
}
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