Received Date: 17 June, 2017 Accepted Date: 22 August, 2017
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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.
Oscillation, Dynamic equation, Time scales, Deviating arguments
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
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
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.
Now we state and prove our main result.
Assume that (H1) and (H2) hold. In addition, assume that rΔ (t) ≥ 0. Then every solution of Equation (1) oscillates if the inequality
and x(t) = y(t) + p(t) y(τ (t)) (3)
has no eventually positive solution.
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)
Multiplying Equation (7) by q(t) , then (8)
Integrating Equation (8) from a to b, we get
Substituting (9) in (4) we obtain,
Since for some
Substituting the last inequality in Equation (10), we get
which is the inequality (2).
As a consequence of this, we have a contradiction and therefore every solution of Equation (1) oscillates.
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).
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).
All the conditions of Theorem (2) are satisfied.
Taking α (s) = s , we see that
Therefore (12) is oscillatory.