OSCILLATORY AND ASYMPTOTIC SOLUTIONS OF FOURTH ORDER NON-LINEAR DIFFERENCE EQUATIONS WITH DELAY

 Ananthan V^1*, Kandasamy S² and Vemuri Lakshminarayana³

¹Assistant Professor, Department of Mathematics, Aarupadai Veedu Institute of Technology, Vinayaka Missions University, Paiyanoor, Kancheepuram-603104, Tamilnadu, India

²Professor, Department of Mathematics, Vinayaka Missions Kirupananda Variyar Engineering College, Vinayaka Missions University, Salem- 636308, Tamilnadu, India

³Principal, Aarupadai Veedu Institute of Technology, Vinayaka Missions University, Paiyanoor, Kancheepuram-603104, Tamilnadu, India.

*Corresponding Author:

 Ananthan V
Assistant Professor, Department
of Mathematics, Aarupadai Veedu
Institute of Technology, Vinayaka
Missions University, Paiyanoor
Kancheepuram-603104, Tamilnadu, India
E-mail: ltamilselvi@avit.ac.in

Received Date: 17 June, 2017 Accepted Date: 22 August, 2017

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Abstract

The objective of this paper is to study the oscillatory and asymptotic solutions of fourth order nonlinear delay difference equation of the form

Example is given to illustrate the results.

Keywords

Difference equations, Asymptotic, Nonlinear, Delay

Introduction

In this paper, we study the oscillatory and asymptotic behavior of solution of fourth order nonlinear delay difference equation of the form

(1)

Here Δ is the forward difference operator and defined by Δy_n=y_n+1–y_n where k is a fixed nonnegative integer and {a_n}, {p_n} and {q_n} are sequence of nonnegative integers with respect to the difference equation (1) throughout. A nontrivial solution {y_n} of equation (1) is said to be oscillatory if for any N ≥ nothere exists n > N such that y_n+1y_n ≤ 0. Otherwise, the solution is said to be non-oscillatory (Agarwal, 1992; Artzrouni, 1985; Cheng and Patula, 1993; Peterson, 1995; Philos and Purnaras, 2001) We shall assume that the following conditions hold:

(c₁) {a_n}, {p_n} and {q_n} are real sequences and an ≤ 0 for infinitely many values of n.

(c₂) f: R→R is continuous and yf(y)>0, for all y ≠ 0.

(c₃) σ (n) ≥ 0 is an integer such that

Main Results

Theorem 1

In addition to the conditions

(c₁), (c₂), (c₃), (c₄), if the conditions are

Then every solution of equation (1) is oscillatory.

Proof

Suppose that the equation (1) has non-oscillatory solution {y_n} is eventually positive. Then there is a positive integer n_o such that y_σ(n) ≥ 0, f or n ≥ n_o implies that {y_n} is non-oscillatory. Without loss of generality we can assume that there exists an integer n₁ ≥ n_o such that

From equation (1) we have

(2)

In view of the conditions

(c₂), (c₃), (H₂) and from the equation (2), we obtain for all n ≥ n₂ (3)

Summing the inequality (3) from n₂ to n —1 we have

for all n ≥ n₂ (4)

Therefore

Then there exists an integer n₂ ≥ n₁ and k₂ >0 such that

(5)

Summing the inequality (5) from n₃ to n —1, we have

(6)

In view of the condition (c₄), and from the inequality (5), we obtain which is a contradiction to the fact that for all large n. This shows that

For all large n.

Let Then L is finite or infinite.

Case 1

L > 0 is finite.

In view of (c₂), (c₃) we have

This implies that

for all n

Then there exists an integer n₄ ≥ n₃ and from equation (1), we obtain

, for all n≥ n₄ (7)

Summing the inequality (7) from n₄ to n —1, we have

for all n ≥ n₄ (8)

In view of (H₂), (H₃) from inequality (8), we find that ∞ ≤ 0, as n→∞ which is a contradiction.

Case II

L=∞

In view of (H₂), there exists an integer n₄ ≥ n₃ and k₃ > 0 such that f(y_σ(n))>k₃, for all n ≥ n₅

Therefore, from equation (1), we obtain

for all n ≥ n₅ (9)

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 (c₁), (c₂), (c₃), (c₄), if the conditions of theorem 1 hold. Then every bounded solution of equation (1) is oscillatory.

Proof

Proceeding as in the proof of theorem 1 with assumption that is {y_n} bounded non-oscillatory solution (1).

Therefore, from inequality (7) of theorem 1, we find that

(10)

By the definition of R_n and from the inequality (10) we find that:

for all n ≥ n₄ (11)

In view of, (H₂), (H₃) and (c₄), we have

for all large n.

This shows that sequence {y_n} is a bounded oscillatory solution of equation (1).

This completes the proof.

Theorem (A):

Let a_n=p_n≡1 and f be non-decreasing.

If then equation (1) has a nonoscillatory solution that approaches a nonzero real number as n→∞.

Asymptotic Behavior

In this section, we obtain a sufficient condition for the asymptotic behavior of solutions of equation (1). We do not require q_n > 0 here. Let A_n, B_n, and C_n be defined by

Theorem 2

Let f(u)be non-decreasing and let d>0 be a constant such that a_n ≥ d for all n ≥ n_o.

Suppose that

Then equation (1) has a bounded non-oscillatory solution that approaches a nonzero limit (Philos, 2005; Philos, 2004; Philos, 2004; Kordonis, 2004; Philos and Purnaras, 2004).

Proof

Let c>0 and let N be so large that

Let the Banach space β_N and the set N μ ⊆ β_N be the same as in theorem (A) and define the operator T: μ→ β_N by

Where

Conclusion

Similar to the proof of theorem (A), we can show that the mapping T satisfies the hypotheses of Schauder’s fixed point theorem (Philos and Purnaras, 2005; Philos and Purnaras, 2004; Julio, 2005; Philos and Purnaras, 2008; Philos and Purnaras, 2010).

Hence, T has a fixed point Yєμ, and it is clear that Y= {y_n}is a non-oscillatory solution of equation (1) for n ≥ Nand has the desired properties.

It should be pointed out that Theorem (A) is actually a special case of the above result. We conclude this paper with a simple example of Theorem (2).

Example:

(13)

Where m is a positive integer. All conditions Theorem (2) are satisfied, so equation (13) has a bounded nonoscillatory solution that approaches a non-zero limit.

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