OSCILLATION OF SECOND ORDER OF NEUTRAL DYNAMIC EQUATION WITH DISTRIBUTED DEVIATING ARGUMENTS

Bhuvaneswari A.K.^1*, Thamizhsudar M¹, Lakshminarayana V²

¹Faculty of Department of Mathematics, Aarupadai Veedu institute of technology Vinayaka Missions University, Paiyanoor-603 104, Chennai, India

²Principal, Aarupadai Veedu institute of technology Vinayaka Missions University, Paiyanoor-603 104, Chennai, India

*Corresponding Author:

Bhuvaneswari A.K.
Faculty of Department of Mathematics
Aarupadai Veedu institute of technology
Vinayaka Missions University, Paiyanoor-603 104
Chennai, India
E-mail: bhuvanaharini73@yahoo.com

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

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Abstract

In this paper, we establish some oscillation criteria for second order neutral dynamic equation with deviating arguments of the form:

On an arbitrary time scale T. An example illustrating the main result is included.

Keywords

Oscillation, Dynamic equation, Time scales, Deviating arguments

Introduction

In a neutral dynamic equation with deviating arguments, the highest order derivative of the unknown function appears with and without deviating arguments. These equations find numerous applications in natural sciences and technology.

In this paper, we study the oscillatory behaviour of second order neutral dynamic equation with distributed deviating arguments of the form

(1)

where 0 < a < b , τ (t) : T→T is right dense continuous function such that τ (t) ≤ t and is right dense continuous function decreasing with respect to ξ, and 0 ≤ p(t) <1 are real valued right dense continuous function defined on T, p(t) is increasing and (H₂): f :T × R→ R is continuous function such that uf (t,u) > 0 for all u ≠ 0 and there exists a positive function q(t) defined on T such that

A non trivial function y(t) is said to be a solution of (1) if

and

for and y(t) satisfies equation (1) for

A non trivial solution of Equation (1) is called oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called non oscillatory (Bohner and Peterson, 2001; Bohner and Peterson, 2003; Bohner and Saker, 2004; Akin, et al., 2007).

We note that if T = R we have σ (t) = t, μ (t) = 0 then equation (1) becomes second order neutral differential equation

If T = N we have σ (t) = t +1, μ (t) =1

then equation (1) becomes

For more papers related to neutral dynamic equations with distributed deviating arguments, we refer the reader to [8,9]. The books [1,2] gives time scale calculus and its applications.

Main Results

Now we state and prove our main result.

Theorem 4.1

Assume that (H₁) and (H₂) hold. In addition, assume that r^Δ (t) ≥ 0. Then every solution of Equation (1) oscillates if the inequality

(2)

where

and x(t) = y(t) + p(t) y(τ (t)) (3)

has no eventually positive solution.

Proof

Let y(t) be a non-oscillatory solution of Equation (1).

Without loss of generality assume that y (t) > 0 for then and for From Equation (1) and (H₂), we have (4)

and is an eventually decreasing function. Now we claim that eventually.

If not, there exists a such that then we have and it follows that (5)

Now integrating Equation (5) from t₂ to t and using (H₁) , we obtain as t →∞, which contradicts the fact that x(t) > 0 for all 0t ≥ t . Hence r (xΔ (t)) is positive.

Therefore there is a (6)

Therefore (7)

Multiplying Equation (7) by q(t) , then (8)

Integrating Equation (8) from a to b, we get

(9)

Substituting (9) in (4) we obtain,

(10)

Since for some

Therefore

Substituting the last inequality in Equation (10), we get

Or

which is the inequality (2).

As a consequence of this, we have a contradiction and therefore every solution of Equation (1) oscillates.

Theorem 2

Assume that (H₁) and (H₂) hold. In addition, assume that is increasing with respect to t and there exists a positive right dense continuous, Δ differentiable function α (t) such that (11)

where and Then every solution of Equation (1) is oscillatory (Higgins, 2008; Saker, 2010; Thandapani, et al., 2011).

Proof

Let y(t) be a non-oscillatory solution of (1).

Without loss of generality assume that y (t) > 0 for t ≥ t₀ , then and

Define the function

Then z(t) > 0 . Now

Integrating from t₇ to t, we obtain

which contradicts Equation (11)

Hence the proof.

Example: Consider the following second order neutral dynamic equation (Candan, 2011; Candan, 2013).

(12)

Conclusion

All the conditions of Theorem (2) are satisfied.

Now

Taking α (s) = s , we see that

Therefore (12) is oscillatory.

References

1. Akin, E., Bohner, M. and Saker, S.H. (2007). Oscillation criteria for a certain class of second order Emden fowler dynamic equations. Electronic Transactions on Numerical Analysis. 27 : 1-12.
2. Bohner, M. and Peterson, A. (2001). Dynamic equations on time scales-An introduction with applications. Boston, Birkhauser, 175 Fifth Avenue, New York, NY10010, USA.
3. Bohner, M. and Peterson, A. (2003). Advances in dynamic equations on timescales”, Boston, Birkhauser, 175 Fifth Avenue, New York, Y10010, USA.
4. Bohner, M, Saker, S.H. (2004). Oscillation criteria for perturbed nonlinear dynamic equations. Mathematical and Computer Modeling. 40 : 249-260.
5. Candan, T. (2011). Oscillation of second order nonlinear neutral dynamic equation on time scales with distributed deviating arguments. Comput. Math. Appl. 62(11) : 4118-4125.
6. Candan, T. (2013). Oscillation criteria for second -order nonlinear neutral dynamic equations with distributed deviating arguments on time scales. Advances in Difference Equations. 1 : 5.
7. Higgins, R.J. (2008). Oscillation theory of dynamic equations on time scales. a Dissertation, University of Nebraska, Lincoln.
8. Saker, S. (2010). Oscillation theory of dynamic equations on time scales-second and third orders. Germany, LAP Lambert Academic publishing, Dudweiler Landstr, 99, 66123 Saarbrucken, Germany.
9. Thandapani, E., Piramanantham V. and Pinelas, S. (2011). Oscilation criteria for second order neutral delay dynamic equation with mixed non-linearities. Advances in Difference Equations. p. 1-14.