Integral Oscillation Criteria for Second-Order Linear neutral Delay Dynamic Equations on Time Scales

In this paper we present several sufficient conditions for oscillation of the second-order linear neutral delay dynamic equation on a time scale T. Our results as a special case when T = R and T = N improve some well-known oscillation results for second-order neutral delay differential and difference equations.


Introduction
In 1988, Stefan Hilger introduced the calculus of measure chain in order to unify continuous and discrete analysis.Berned Aulbach, who supervised Stefan Hilger's Ph.D. thesis (Hilger, S., 1990, p18-56), points out the three main purposes of this new calculus: Unification -Extension -Discretization.
For many purposes in analysis it is sufficient to consider a special case of a measure chain, a so-called time scale, which simply is a closed subset of the real numbers.We denote a time scale by the symbol T. The two most popular examples are T = R and T = Z that represent the classical theories of differential and of difference equations.Since Stefan Hilger formed the definition of derivatives and integrals on time scales, several authors has expounded on various aspects of this new theory, see the paper by (Agarwal et al., 2002, p1-26) and the references cited therein.The books on the subject of time scales, i.e., measure chain, by Bohner andPeterson (2001, 2003) summarize and organize much of time scale calculus.
In this paper, we are concerned with the oscillation of the second-order linear dynamic equation (y(t) + p(t)y(t − τ)) ∆∆ + q(t)y(t − δ) = 0 (1) on a time scale T.
Since we are interested in asymptotic behavior of solutions, we will suppose that the time scales T under consideration is not bounded above; i.e., it is a time scale interval of the form [t 0 , ∞) T = [t 0 , ∞) ∩ T.
By a solution of equation ( 1 1) for all t > t y .Our attention is restricted to those solutions of equation ( 1) which exist on some half line [t y , ∞) and satisfy sup{|y(t)| : t > t 1 } > 0 for any t 1 > t y .A solution y(t) of equation ( 1) is said to be oscillatory if it neither eventually positive nor eventually negative.Otherwise it is called nonoscillatory.The equation itself is called oscillatory if all its solutions are oscillatory.
Moreover, we intend to use the Riccati integral equations and the theory of integral inequalities (Kwong Man Kam, 2006, p1-18) for obtaining several oscillation criteria for (1).Hence the paper is organized as follows: In section 2, we present some preliminaries on time scales.In section 3, we establish some new sufficient conditions for oscillation of (1).

Some preliminaries on time scales
A time scale T is an arbitrary nonempty closed subset of the real numbers R. On any time scale T, we define the forward and backward jump operators by The set of functions f : T → R that are differentiable and whose derivative is rd-continuous function is denoted by A function p : T → R is called positively regressive (we write p ∈ R + ) if it is rd-continuous function and satisfies 1 + µ(t)p(t) > 0 for all t ∈ T.
For a function f : T → R (the range R of f may be actually replaced by any Banach space) the (delta) derivative is defined by if f is continuous at t and t is right-scattered.If t is right-dense then the derivative is defined by provided this limit exists.
A function f : [a, b] → R is said to be differentiable if its derivative exists, and a useful formula is We will make use of the following product and quotient rules for the derivative of the product f g and the quotient f /g ( where gg σ 0) of two differentiable functions f and g For a, b ∈ T and a differentiable function f , the Cauchy integral of f ∆ is defined by and infinite integral is defined as An integration by parts formula reads

Main results
Before stating our main results in this paper, we start with the following lemmas.
Lemma 1 Assume that (H3) hold and the inequality has a positive solution x on [t 0 , ∞) T .Then there exists a T ∈ [t 0 , ∞) T , sufficiently large, so that x ∆ (t) ≥ 0 and x ∆ (t) proof.The proof is similar to the proof of Lemma 1 in (Erbe, L. ,2006, p65-78).
proof.Suppose to the contrary that y(t) is a nonoscillatory solution of equation ( 1).Without loss of generality, we may assume that y(t) is an eventually positive solution of (1) with y(t − N) > 0 where N = max{τ, δ} for all t > t 0 sufficiently large.We shall consider only this case, since the substitution z(t) = −y(t) transform Eq. ( 1) into an equation of the same form.Set From ( 10) and (1) we have for all t > t 0 , and so x ∆ (t) is an eventually decreasing function.We first show that x ∆ (t) is eventually nonnegative.Indeed, since q(t) is a positive function, the deceasing function x ∆ (t) is either eventually positive or eventually negative.Suppose there exists an integer t 1 ≥ t 0 such that x ∆ (t 1 ) = c < 0, then x ∆ (t) < x ∆ (t 1 ) = c for t ≥ t 1 , hence x ∆ (t) ≤ c, which implies that which contradicts the fact that x(t) > 0 for all t > t 1 .Hence x ∆ (t) is eventually nonnegative.Therefore, we see that there is some t 1 such that This implies that Then, for t ≥ t 1 = t 0 + δ sufficiently large, we see that From ( 11) and ( 13) we obtain for t ≥ t 1 Then from ( 14), we have ¢ www.ccsenet.org

Journal of Mathematics Research
February, 2010 Integrating the above inequality from t 1 to t, we get In view of (12), we obtain From ( 16) and ( 17) we get Therefore, x(σ(s) − δ) Let Then from ( 18), ( 19), it is easy to see that Then, by lemma 1, for sufficiently large t, there exists β(t) ≥ t such that 1 Since lim t→∞ β(t) α(t) = 0 we have which contradicts the condition (9).The proof is complete.
proof.Suppose to the contrary that y(t) is a nonoscillatory solution of Eq. ( 1) and let t 1 ≥ t 0 be such that y(t) 0 for all t ≥ t 1 , so without loss of generality, we may assume that y(t) is an eventually positive solution of Eq. (1).From (15), we get We proceed as in the proof of Theorem 1 and it follows that which contradicts the condition ( 21).The proof is complete.
proof.We proceed as in the proof of Theorem 1 to prove that there exists t 1 ≥ t 0 such that (20) holds for t ≥ t 1 .From ( 20), it follows that which contradicts the condition (23).
proof.We proceed as in the proof of Theorem 1 to prove that there exists t 1 ≥ t 0 such that (20) holds for t ≥ t 1 .From (20), it follows that which contradicts the condition (24).
proof.Suppose that Eq. ( 1) has a nonoscillatory solution y(t).We may assume without loss of generality that y(t) > 0 for all t > t 0 .We will consider only this case, since the proof when y(t) is eventually negative is similar.In view of Lemma 2, for each positive constant k ∈ (0, 1), there exists a t 1 = max{t k , t 0 } such that ¢ www.ccsenet.org

Journal of Mathematics Research
February, 2010 From ( 15) and from ( 26), we get We proceed as in the proof of Theorem 1, so we get From lemma 1, for sufficiently large t, there exists β(t) ≥ t such that 1 β(t) ≤ x ∆ (σ(t)) x(σ(t)) ≤ 1 t .Hence Since lim t→∞ which contradicts the condition ( 25).The proof is complete.
Assume that the condition (23) fails, and In this case we have the following result.
proof.Assume that Eq. ( 1) has a positive solution y(t) for all t ≥ t 1 .Then from condition (31) we have, From lemma 1, for sufficiently large t, we have Then from ( 32) and ( 33) we get Let ¢ www.ccsenet.org/jmrISSN: 1916-9795 For that, From ( 20), ( 34) and ( 35) we get From ( 36), for sufficiently large t, we have which is a contradiction.This complete the proof.
Remark 1 From Theorem 1 and Theorem 2 we can obtain different conditions for oscillation of Eq. (1) by choosing α(t) = t.
The following theorem gives Philos-type oscillation criteria for Eq.(1).First, let us introduce now the class of functions R which will be extensively used in the sequel.

Journal of Mathematics Research
February, 2010 proof.Suppose to the contrary that y(t) is a nonoscillatory solution of Eq. ( 1) and let t ≥ t 1 be such that y(t) 0 for all t ≥ t 1 , so without loss of generality, we may assume that y(t) is an eventually positive solution of Eq. ( 1) with y(t − N) > 0 where N = max{τ, δ} for all t ≥ t 1 sufficiently large.We proceed as in the proof of Theorem 1. From (15) we get x ∆ (σ(s))α ∆ (s)H(t, s) x(σ(s) − δ) ∆s Then by using ( 17) we get where H(t, t) = 0. Therefore by using Lemma 3, with we get that So x(t 1 − δ) .
proof.By proceeding as in the proof of Theorem 5 and from (41) we get 1 H(t, t 1 ) which contradicts the condition (42).Then every solution of Eq. ( 1) oscillates.
Remark 2 With an appropriate choice of the functions H ∈ C rd (D, R) and h ∈ C rd (D 0 , R).We can take H(t, s) = (t − s) m , (t, s) ∈ D with m > 1.It is clear that H belongs to the class R. Now, we claim that proof.
We consider the following two case: Case Using Hardy et al. inequality (Hardy, 1952) x m − y m ≥ my m−1 (x − y) for all x ≥ y > 0 and m ≥ 1 (46) we have Then from ( 45) and (47), we have and this proves (44).
From the above claim and Theorem 5, we have the following Kamenev-type oscillation criteria for Eq.(1).
Since lim t→∞ β(t) α(t) = 0 we have which is contradicts (49), and consequently, Eq. ( 1) has no eventually positive solution.Similarly, by using the same technique we can prove that Eq. ( 1) has no eventually negative solution.Thus Eq. ( 1) is oscillatory.
T → R is called rd-continuous provided it is continuous at right-dense points in T and its left-sided limits exist (finite) at left-dense points in T. The set of rd-continuous functions right-dense if t < supT and σ(t) = t, left-scattered if ρ(t) < t and right-scattered if σ(t) > t.The graininess function µ for a time scale T is defined by µ(t) = σ(t) − t.