Nonlinear Oscillation and Asymptotic Behavior of Second-order Neutral Dynamic Equations on Time Scales

The paper is to study the oscillation and asymptotic behavior of the second-order nonlinear neutral dynamic equation   ( ) | ( ) | sgn ( ) ( ) | ( ) | sgn ( ) 0 r t y t y t q t x t x t        on an arbitrary time scale T , where 1   , 0   are constants, , ( , (0, )) rd r q C   T , ( ) : y t  ( ) ( ) ( ( )) x t p t x t   , ( ,[0,1]) rd p C  T ,   ( , ) rd C T T , ( ) t t   for tT and lim ( ) t t     . By using a generalized Riccati transformation technique, we obtain some sufficient conditions which ensure that every solution of the equation oscillates or converges to zero. Our results improve and extend some existing results in which ( ) 0 p t  and  ,  are quotients of odd positive integers.

Recall that a solution of (1) is a nontrivial real function x such that 1 ( ) ( ) ( ( )) [ , )  1) is said to be oscillatory if all its solutions are oscillatory.
The concept of dynamic equations on time scales was introduced by Hilger in his PhD thesis (Hilger, 1990) with the motivation of providing a unified approach to continuous and discrete calculus.Thus, the notion of a generalized delta derivative ( ) f t  was introduced, where the domain of the function f is a so-called "time scale" T (an arbitrary nonempty closed subset of ).If the time scale T is the real numbers , then the usual derivative is retrieved, that is, . On the other hand, if the time scale T is taken to be the integers , then the generalized delta derivative reduces to the usual forward difference, that is, . Not only can the theory of dynamic equations on time scales unify the theories of differential equations and difference equations, but it is also able to extend these classical cases to cases "in between," e.g., to the so-called q -difference equations.For an introduction to time scale calculus and dynamic equations, we refer to the seminal book by Bohner and Peterson (2001).For advances in dynamic equations on time scales, one can see the book by Bohner and Peterson (2003).Throughout the paper it is assumed that the reader is familiar with time scale calculus.
Recently, Saker (2005) established some oscillation criteria for (1) when ( ) 0 p t  and 1     is an odd positive integer.Hassan (2008) obtained some sufficient conditions for the oscillation of (1) when ( ) 0 p t  and    is a quotient of odd positive integers.Hassan (2008) improved and extended the results of Saker (2005).Grace et al. (2008Grace et al. ( , 2009) ) gave some new oscillation results for (1) when ( ) 0 p t  and  ,  are quotients of odd positive integers.
It is easy to see that the cases considered in (Saker, 2005;Hassan, 2008;Grace et al., 2008;Grace et al., 2009) only are some special cases of (1) and that all the results of (Saker, 2005;Hassan, 2008;Grace et al., 2008;Grace et al., 2009) can not be applied to (1) when ( ) p t is not identically equal to zero or  ,  are not equal to quotients of odd positive integers.Accordingly, it is of great interest to study the oscillation and asymptotic behavior of (1) when ( ) p t is not identically equal to zero and  , 0   are constants.In this paper, we will establish some new oscillation criteria for (1) when 0 ( ) 1 p t   for t T and 1   , 0   are constants.We don't restrict    , which was used in (Saker, 2005) and (Hassan, 2008).Our results improve and extend some of those in (Saker, 2005;Hassan, 2008;Grace et al., 2008;Grace et al., 2009).
The following lemma will play an important role in the proof of our main results.
Lemma 1. ((Bohner and Peterson, 2001), p. 32, Theorem 1.87) Let : In what follows, for convenience, when we write a functional inequality without specifying its domain of validity we assume that it holds for all sufficiently large t .

Main results
In this section, we will present and prove our main results.We will consider both the case when holds and the case when holds.
Theorem 1. Suppose that (3) and (4) hold.Furthermore, assume that there exists a positive delta differentiable function  such that for all 4 3 0 where 3 1/ ( ) : ( )  1) is oscillatory.Proof.Assume that x is a nonoscillatory solution of (1).Without loss of generality, we may assume that x is an eventually positive solution of (1).Then it follows from (2) and (3) that there exists 1   .This contradicts (7).Thus, we get that (9) holds.From ( 2) and ( 7 From ( 8), ( 9) and ( 12), there exists 3 2 [ , ) for the delta derivatives of the product fg and the quotient / f g of differentiable functions f and g , where  is the forward jump operator on T , : Hence, from ( 13) , ( 14) and the last equality we obtain

y t y t y t y t y t y t y t
Hence, from ( 15) and ( 17) we find (1 ) From ( 14) we get Then from ( 19) we obtain Next, we consider the following three cases: Case (i).Let   where  is defined as in Theorem 1.


, which implies a contradiction to (6).The proof is complete.
Theorem 2. Suppose that (3) and ( 5) hold.Furthermore, assume that there exists a positive delta differentiable function  such that (6) holds and that there exists a constant 0   such that the following conditions hold: 1 (1 ) ( ) 0 p t     for t T Then every solution of ( 1) is oscillatory or converges to zero as t   .Proof.Assume that x is a nonoscillatory solution of (1).Without loss of generality, we may assume that x is an eventually positive solution of (1).Then there exists 1 0 t t  such that ( 7) and ( 8) hold.Therefore, we see that ( ) | ( ) | sgn ( ) r t y t y t positive constants,  is the forward jump operator on T and : Integrating both sides of the last inequality from 4 t to t , we obtain for