ISSN (0970-2083)

All submissions of the EM system will be redirected to Online Manuscript Submission System. Authors are requested to submit articles directly to Online Manuscript Submission System of respective journal.

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

Ananthan V1*, 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
Institute of Technology, Vinayaka
Missions University, Paiyanoor
E-mail: ltamilselvi@avit.ac.in

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

Visit for more related articles at Journal of Industrial Pollution Control

## 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 Δyn=yn+1–yn where k is a fixed nonnegative integer and {an}, {pn} and {qn} are sequence of nonnegative integers with respect to the difference equation (1) throughout. A nontrivial solution {yn} of equation (1) is said to be oscillatory if for any N ≥ nothere exists n > N such that yn+1yn ≤ 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) {an}, {pn} and {qn} 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

(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 {yn} is eventually positive. Then there is a positive integer no such that yσ(n) ≥ 0, f or n ≥ no implies that {yn} is non-oscillatory. Without loss of generality we can assume that there exists an integer n1 ≥ no 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(yσ(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 {yn} is oscillatory.

In fact yn < 0, yn-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 {yn} bounded non-oscillatory solution (1).

Therefore, from inequality (7) of theorem 1, we find that (10)

By the definition of Rn 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 {yn} is a bounded oscillatory solution of equation (1).

This completes the proof.

Theorem (A):

Let an=pn≡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 qn > 0 here. Let An, Bn, and Cn be defined by Theorem 2

Let f(u)be non-decreasing and let d>0 be a constant such that an ≥ d for all n ≥ no.

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 βN and the set N μ ⊆ βN be the same as in theorem (A) and define the operator T: μ→ βN 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 YÃÂμ, and it is clear that Y= {yn}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.

#### References

1. Agarwal, R.P. (1992). Difference equations and inqualities, Marcel Dekker, New York, USA.
2. Artzrouni, M. (1985). Generalized stable population theory. J. Math.Biology. 21 : 363-381.
3. Cheng, S.S. and Patula, W.T. (1993). An existence theorem for a nonlinear difference equation. Nonlinear Anal. 20 : 193-203.
4. Julio, G.D, Philos, C.G. and Purnaras, I.K. (2005). An asymptotic property of solutions to linear nonautonomous delay differential equations. Electron. J. Differential Equations. 10 : 1-9.
5. Kordonis, I.G.E., Philos, C.H.G. and Purnaras, I.K. (2004). On the behavior of solutions of linear neutral integrodifferential equations with unbounded delay. Georgian Math. J. 11 : 337-348
6. Peterson, A. (1995). Sturmian theory and oscillation of a third order linear difference equation, in: Boundary value problems for functional differential equations. World Sci.Pub.,River Edge, Nj. 261-267
7. Philos, C.H.G. and Purnaras, I.K. (2001). Periodic first order linear neutral delay differential equations. Appl. Math. Comput.117 : 203-222.
8. Philos, C.H.G., Purnaras, I.K. and Sficas, Y.G. (2005). On the behaviour of the oscillatory solutions of second order linear unstable type delay differential equations. Proc. Edinburgh Math. Soc. 48 : 485-498.
9. Philos, C.H.G., Purnaras, I.K. and Sficas, Y.G. (2004). On the behavior of the oscillatory solutions of first or second order delay differential equations. J. Math. Anal. Appl. 291 : 764-774.
10. Philos, C.H.G. and Purnaras, I.K. (2004). The behavior of solutions of linear Volterra difference equations with infinite delay. Comput. Math. Appl. 47 : 1555-1563.
11. Philos, C.H.G. and Purnaras, I.K. (2004). Asymptotic properties, nonoscillation, and stability for scalar first order linear autonomous neutral delay differential equations. Electron. J. Differential Equations. 03 : 1-17.
12. Philos, C.H.G. and Purnaras, I.K. (2005). The behavior of the solutions of periodic linear neutral delay difference equations. J. Comput. Appl. Math. 175 : 209-230.
13. Philos, C.H.G. and Purnaras, I.K. (2004). An asymptotic result for some delay difference equations with continuous variable. Advances in Difference Equations. 1 : 1-10.
14. Philos, C.H.G. and Purnaras, I.K. (2008). Sufficient conditions for the oscillation of linear difference equations with variable delay. J. Difference Equ. Appl. 14 : 629-644.
15. Philos, C.H.G. and Purnaras, I.K. (2010). An asymptotic result for second order linear non-autonomous neutral delay differential equations. Hiroshima Math. J. 40 : 47-63.