OSCILLATION CRITERIA OF SECOND ORDER NEUTRAL DELAY DYNAMIC EQUATIONS WITH DISTRIBUTED DEVIATING ARGUMENTS

In this paper we establish some oscillation theorems for second order neutral dynamic equations with distributed deviating arguments. We use the Riccati transformation technique to obtain sufficient conditions for the oscillation of all solutions. Further, some examples are provided to illustrate the results.


Introduction
In this paper we are concerned with the oscillatory behavior of solutions of second order neutral type dynamic equations with distributed deviating arguments of the form (r(t)(x(t) + p(t)x(t − τ )) ∆ ) ∆ + b a q(t, ξ)f (x(g(t, ξ)))∆ξ = 0, t ∈ T (1.1) subject to the conditions: (A 1 ) r(t), p(t) are positive real valued rd-continuous functions on time scales with 0 ≤ p(t) < 1; (A 2 ) f ∈ C(R, R) such that uf (u) > 0 for u = 0, and f (−u) = −f (u); (A 3 ) g(t, ξ) ∈ C rd (T × [a, b] T , T), g(t, ξ) ≤ t, ξ ∈ [a, b] T ,where [a, b] T = {t ∈ T : a ≤ t ≤ b}, g is strictly increasing with respect to t and decreasing with respect to ξ, and the integral of equation (1.1) is in the sense of Riemann (see [7]).
By a solution of equation (1.1), we mean a nontrivial real valued function x(t) which has the properties x(t) + p(t)x(t − τ ) ∈ C 1 rd ([t y , ∞) T and r(t)[x(t) + p(t)x(t − τ )] ∆ ∈ C 1 rd ([t y , ∞) T and satisfying equation (1.1) for all t ∈ [t 0 , ∞) T .We restrict our attention to nontrivial solutions of equation (1.1) that exist on some half-line [t y , ∞) T , and satisfying sup{| x(t) |: t ∈ [t 1 , ∞) T } > 0 for any t 1 ∈ [t y , ∞) T .A solution x(t) of equation (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is called nonoscillatory.The equation itself is called oscillatory if all its solutions are oscillatory.Since we are interested in oscillatory behavior of solutions, we will suppose that the time scale T under considerations is not bounded above, that is, it is a time scale interval of the form [t 0 , ∞) T .
We note that if T = R, then we have f ∆ (t) = f ′ (t), and equation (1.1) becomes the second order neutral differential equation with distributed deviating arguments of the form (r(t)(x(t) + p(t) , and the equations (1.1) becomes the second order neutral difference equation with distributed deviating arguments of the form Recently there has been an increasing interest in studying the oscillation of solutions of dynamic equations with continuous deviating arguments, see for example [1,6,14,[16][17][18] and the references cited therein.To the best of our knowledge no paper has been published in dynamic equations with distributed deviating arguments.This motivated us to study the oscillatory behavior of equation (1.1).
The purpose of this paper is to derive some sufficient conditions for the solutions of the equation (1.1) to be oscillatory under the conditions In Section 2, we present some basic lemmas , and in Section 3 we will use the Riccati transformation technique to prove our oscillation results of the equations (1.1) under the condition (C 1 ).Also we derive sufficient condition for the equation (1.1) to be oscillatory under the condition (C 2 ).In Section 4, we present some examples to illustrate our main results.

Some basic lemmas
In this section, we give some preliminary lemmas which are useful to prove the main results.
Lemma 2.1.[4]Assume that v ∈ T → R is strictly increasing, and Proof.The proof is obvious. Hence ∆s which implies by condition (C 1 ) that y(t) → −∞ as t → ∞.This contradicts the fact that y(t) > 0 for all t ∈ [t i , ∞) T .Hence r(t)y ∆ (t) > 0 eventually.

Oscillation results
In this section, first we derive some sufficient conditions for the solutions of equation (1.1) to be oscillatory when the condition (C 1 ) holds.We begin with the following theorem.
Theorem 3.1.Assume that condition (C 1 ) holds, and further assume that there exist Without loss of generality we may assume that x(t) is an eventually positive solution of equation (1.1).Then there is a t 1 ≥ t 0 such that x(t) > 0, x(t−τ ) > 0, and Then using Lemma 2.3, there exists a t 2 ≥ t 0 such that From Lemma 2.3 and using , and we write Using the fact that g(t, ξ) is decreasing with respect to ξ, we have from Then clearly w(t) ≥ 0 for all t ∈ [t 2 , ∞) T , and Let g b (t) = g(t, b).Then using Lemma 2.1, we have From (3.6) and (3.7), we obtain Since y ∆ (t) ≥ 0, and (r(t)y ∆ (t)) ∆ ≤ 0, t ≥ t 1 , we have From (3.8) and (3.9), we obtain (3.10) Multiplying (3.10) by H(t, s) and then integrating from T to t , for any t ≥ T ≥ t 2 , we have Using integrating by parts, we have where M is a constant, which contradicts (3.1).Suppose that x(t) is an eventually negative solution of equation (1.1).Then by taking z(t) = −x(t), we have that z(t) is eventually positive solution of the equation (1.1), since f (−u) = −f (u).Similar to the proof as above we obtain a contradiction.This completes the proof.
As a consequence of Theorem 3.1, we obtain the following corollary.
Then every solution of equation (1.1) is oscillatory.

then every solution of equation (1.1) is oscillatory.
Next, if we consider α(t) = t and α(t) = 1 for t ≥ t 0 in Corollary 3.4, we can obtain few more oscillation criteria as corollaries of Theorem 3.1.then every solution of equation (1.1) is oscillatory.
Theorem 3.8.Assume that the hypotheses of Theorem 3.1 hold.Further assume that and there exists a positive delta differentiable function α(t) such that hold.If there exists a function ϕ and for every T ≥ t 0 , where ϕ + (t) = max{ϕ(t), 0}, then every solution of equation (1.1) is oscillatory.
Proof.On the contrary, we assume that (1.1) has a nonoscillatory solution x(t).We suppose without loss of generality that x(t) > 0 for all t ∈ [t 0 , ∞) T .Proceeding as in the proof of Theorem 3.1, for t > u ≥ t 1 ≥ t 0 , we have EJQTDE, 2010 No. 61, p. 8 Let t → ∞ and taking the upper limit, we have lim sup From (3.19), we have w(u) ≥ ϕ(u) for all u ≥ t 0 , (3.20) and lim inf where M is a constant.Let and Suppose to the contrary that On the other hand by (3.26) for any positive number µ > 0, there exists EJQTDE, 2010 No. 61, p. 9 So, for all t ≥ T > t 1 , integration by parts yields that From (3.24), there is a sequence Thus there exist constants N 1 and M such that Then for any ǫ ∈ (0, 1), there exists a positive integer N 2 such that EJQTDE, 2010 No. 61, p. 10 or From (3.36) and (3.37),we have On the other hand, by Schwarz inequality,we have for all large n.In view of (3.31), we obtain  Mc 0 q(s, ξ)[1 − p(g(s, ξ))]π(g(s, ξ))∆ξ∆s ≤ π(t 2 )r(t 2 )y ∆ (t 2 ) + y(t 2 ).
This contradicts (3.49) as t → ∞.Hence every solution of equation (1.1) is oscillatory.The proof is now complete.