Oscillation Properties for Second-Order Half-Linear Dynamic Equations on Time Scales

Since Hilger (1990) introduced the theory of time scales, many authors have expounded on various aspects of this new theory; see the books (Bohner & Peterson, 2001, 2003) and the papers (Agarwal et al., 2007; Bohner & Saker, 2004; Chen, 2010; Chen & Liu, 2008; Došlý & Hilger 2002; Erbe et al., 2008; Hassan, 2008; Hassan, 2009; Karpuz, 2009; Medico & Kong, 2004; Saker, 2005; Zhang, 2011). A time scale T is an arbitrary nonempty closed subset of the reals R (see Hilger, 1990; Bohner & Peterson, 2001, 2003), and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential equations and of difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice–once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a time scale. In this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. Therefore, not only can the theory of dynamic equations 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. Dynamic equations on time scales have a lot of applications in population dynamics, quantum mechanics, electrical engineering, neural networks, heat transfer, and combinatorics. Bohner and Peterson (2001) summarizes and organizes much of time scale calculus. For advances of dynamic equations on time scales, we refer the reader to (Bohner & Peterson, 2003).


Introduction
Since Hilger (1990) introduced the theory of time scales, many authors have expounded on various aspects of this new theory; see the books (Bohner & Peterson, 2001, 2003) and the papers (Agarwal et al., 2007;Bohner & Saker, 2004;Chen, 2010;Chen & Liu, 2008;Došlý & Hilger 2002;Erbe et al., 2008;Hassan, 2008;Hassan, 2009;Karpuz, 2009;Medico & Kong, 2004;Saker, 2005;Zhang, 2011).A time scale T is an arbitrary nonempty closed subset of the reals R (see Hilger, 1990;Bohner & Peterson, 2001, 2003), and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential equations and of difference equations.Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts.The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice-once for differential equations and once again for difference equations.The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a time scale.In this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained.Therefore, not only can the theory of dynamic equations 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.Dynamic equations on time scales have a lot of applications in population dynamics, quantum mechanics, electrical engineering, neural networks, heat transfer, and combinatorics.Bohner and Peterson (2001) summarizes and organizes much of time scale calculus.For advances of dynamic equations on time scales, we refer the reader to (Bohner & Peterson, 2003).
In recent years, there has been a large number of papers devoted to the oscillation and asymptotic behavior of dynamic equations on time scales, and we refer to (Karpuz, 2009;Hassan, 2009;Chen, 2010;Chen & Liu, 2008;Medico & Kong, 2004;Bohner & Saker, 2004;Došlý & Hilger 2002;Saker, 2005;Agarwal et al., 2007;Hassan, 2008;Erbe et al., 2008) and the references cited therein.For the second-order half-linear dynamic equations on an arbitrary time scale T, where γ > 1 is an odd positive integer, a and q are positive rd-continuous functions defined on the time scale interval [t 0 , ∞), Saker (2005) obtained several oscillation criteria.
Later, Agarwal et al. (2007) supposed that γ > 1 is a quotient of odd positive integers and got several sufficient conditions for the oscillation of all solutions of (1).Agarwal et al. (2007) improved and extended the results of Saker (2005).
Very recently, Hassan (2008) supposed that γ > 0 is a quotient of odd positive integers and established some oscillation criteria of (1).Hassan (2008) improved and extended the results of Saker (2005) and Agarwal et al. (2007).
However, the results of Saker (2005), Agarwal et al. (2007) and Hassan (2008) cannot be applied to the following secondorder half-linear dynamic equations on an arbitrary time scale T, where γ > 0 is a constant, a and q are positive rd-continuous functions defined on the time scale interval [t 0 , ∞).Therefore, it is of great interest to study the oscillation of (2) when γ > 0 is a constant.
In this paper, we establish some oscillation criteria for (2) by applying a generalized Riccati substitution, the Pötzsche chain rule and a Hardy-Littlewood-Pólya inequality.Our results improve and extend the results of Saker (2005), Agarwal et al. (2007) and Hassan (2008).Some examples are shown to illustrate our main results.
Since we are interested in the oscillation of solutions near infinity, we assume that sup T = ∞.By a solution of (2) we mean a nontrivial real function Our attention is restricted to those solutions of (2) which exist on the half-line [t x , ∞) and satisfy sup{|x(t)| : t > t * } > 0 for any t * ≥ t x .A solution x of ( 2) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is nonoscillatory.Equation ( 2) is said to be oscillatory if all its solutions are oscillatory.
We shall need the following lemma to prove our main results.
Lemma 1.1 (Hardy et al., 1988) If X and Y are nonnegative, then where the equality holds if and only if X = Y.
The following theorem gives a Philos-type oscillation criterion for (2).
Proof: Assume that x is a nonoscillatory solution of (2).Without loss of generality, assume that x is an eventually positive solution of (2).Define again the function w by ( 9).Proceeding as in the proof of Theorem 2.1, we see that ( 17 where V(s) := γη(s)a − 1 γ (s)(η σ (s)) − γ+1 γ .Applying the integration by parts formula where h + (t, s) is defined as in Theorem 2.2.Take λ = γ+1 γ and define X ≥ 0 and Y ≥ 0 by Therefore, we find which implies a contradiction to (18).Thus, this completes the proof.
Next, we consider the case when holds.It is clear that (24) implies that (3) does not hold.
Theorem 2.3 Suppose that (24) holds.Let η be defined as in Theorem 2.1 such that (4) holds.Furthermore, assume that for every constant C ≥ t 0 , Then every solution of ( 2) is oscillatory or converges to zero as t → ∞.
Proof: Assume that x is a nonoscillatory solution of (2).Without loss of generality, assume that x is an eventually positive solution of (2).Define again the function w by (9).There are two cases for the sign of x ∆ (t).The proof when x ∆ (t) is eventually positive is similar to that of Theorem 2.1 and hence is omitted.