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** Ananthan V ^{1*}, Kandasamy S^{2} and Vemuri Lakshminarayana^{3}**

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

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

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

- *Corresponding Author:

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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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.

Difference equations, Asymptotic, Nonlinear, Delay

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+1}y_{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:

(c_{1}) {a_{n}}, {p_{n}} and {q_{n}} are real sequences and an ≤ 0 for infinitely many values of n.

(c_{2}) f: R→R is continuous and yf(y)>0, for all y ≠ 0.

(c_{3}) σ (n) ≥ 0 is an integer such that

**Theorem 1**

In addition to the conditions

(c_{1}), (c_{2}), (c_{3}), (c_{4}), 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 n_{1} ≥ n_{o} such that

From equation (1) we have

(2)

In view of the conditions

(c_{2}), (c_{3}), (H_{2}) and from the equation (2), we obtain for all n ≥ n_{2} (3)

Summing the inequality (3) from n_{2} to n —1 we have

for all n ≥ n_{2} (4)

Therefore

Then there exists an integer n_{2} ≥ n_{1} and k_{2} >0 such that

(5)

Summing the inequality (5) from n_{3} to n —1, we have

(6)

In view of the condition (c_{4}), 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 (c_{2}), (c_{3}) we have

This implies that

for all n

Then there exists an integer n_{4} ≥ n_{3} and from equation (1), we obtain

, for all n≥ n_{4} (7)

Summing the inequality (7) from n_{4} to n —1, we have

for all n ≥ n_{4} (8)

In view of (H_{2}), (H_{3}) from inequality (8), we find that ∞ ≤ 0, as n→∞ which is a contradiction.

**Case II**

L=∞

In view of (H_{2}), there exists an integer n_{4} ≥ n_{3} and k_{3} > 0 such that f(y_{σ(n)})>k_{3}, for all n ≥ n_{5}

Therefore, from equation (1), we obtain

for all n ≥ n_{5} (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 (c_{1}), (c_{2}), (c_{3}), (c_{4}), 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 ≥ n_{4} (11)

In view of, (H_{2}), (H_{3}) and (c_{4}), 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→∞.

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

Where

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 YÃÂμ, 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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