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.

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

Visit for more related articles at Journal of Industrial Pollution Control

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

equation (1)

where 0 < a < b , τ (t) : T→T is right dense continuous function such that τ (t) ≤ t and equation is right dense continuous function decreasing with respect to ξ, equation and 0 ≤ p(t) <1 are real valued right dense continuous function defined on T, p(t) is increasing and equation (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 equation

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

equation

and

equation

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

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

equation

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

then equation (1) becomes

equation

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 rΔ (t) ≥ 0. Then every solution of Equation (1) oscillates if the inequality

equation (2)

where equation

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 forequation thenequation andequation for equation From Equation (1) and (H2), we have equation (4)

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

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

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

Therefore there is a equationequation (6)

equation

Therefore equation (7)

equation

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

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

equation (9)

Substituting (9) in (4) we obtain,

equation (10)

Since equation for someequation

equation

Therefore

equation

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

equation

Or

equation

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 equation is increasing with respect to t and there exists a positive right dense continuous, Δ differentiable function α (t) such thatequation (11)

where equationand equation 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 tt0 , then equation andequation

Define the function equation

Then z(t) > 0 . Now

equation

Integrating from t7 to t, we obtain

equation

which contradicts Equation (11)

Hence the proof.

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

equation (12)

Conclusion

All the conditions of Theorem (2) are satisfied.

Now equation

Taking α (s) = s , we see that

equation

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.

Copyright © 2024 Research and Reviews, All Rights Reserved