Oscillation theorems for certain third order nonlinear delay dynamic equations on time scales

In this paper, we establish some new oscillation criteria for the third order nonlinear delay dynamic equations


Introduction
In this paper, we are concerned with the oscillation criteria for the following certain third order nonlinear delay dynamic equations b(t) [a(t)(x ∆ (t)) α1 ] ∆ α2 ∆ + q(t)x α3 (τ (t)) = 0 (1.1) on a time scale T. Throughout this paper and without further mention, we assume that the following conditions are satisfied: (C 1 ) α i are ratios of positive odd integers, i = 1, 2, 3; (C 2 ) b, a and q are positive, real-valued, rd-continuous functions defined on T and (1.2) (C 3 ) τ : T → T, τ (t) ≤ t, τ ∆ (t) > 0 for all t ∈ T and τ (t) → ∞ as t → ∞.
The theory of time scales, which has recently received a lot of attention, was originally introduced by Stefan Hilger [1] in his Ph.D. Thesis in 1988, in order to unify, extend and generalize continuous and discrete analysis.The book on the subject of time scales by Bohner and Peterson [2] summarizes and organizes much of time scale calculus and many applications.In recent years, there has been increasing interest in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of various equations on time scales, and we refer the reader to the papers [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].
A time scale T is an arbitrary nonempty closed subset of the real numbers R and since we are interested in the oscillatory behavior of solutions near infinity, we make the assumption throughout this paper that the time scale T is unbounded above.We assume t 0 ∈ T and it is convenient to assume t 0 > 0. We define the time scale interval [t 0 , ∞) T by [t 0 , ∞) T = [t 0 , ∞) T. We assume throughout that T has the topology that it inherits from the standard topology on the real numbers R. The forward and the backward jump operators are defined by: σ(t) = inf {s ∈ T : s > t} and ρ(t) = sup {s ∈ T : s < t}, where inf ∅ = sup T and sup ∅ = inf T. A point t ∈ T is said to be left-dense if ρ(t) = t and t > inf T, right-dense if σ(t) = t, left-scattered if ρ(t) < t and right-scattered if σ(t) > t.A function g : T → R is said to be rd-continuous provided g is continuous at right-dense points and at left-dense points in T, left-hand limits exist and are finite.The set of all such rd-continuous functions is denoted by C rd (T).The graininess function µ for a time scale T is defined by µ(t) = σ(t) − t and for any function f : T → R, the notation f σ denotes f • σ.
By a solution of Eq. (1.1), we mean a nontrivial real-valued function T and satisfies Eq. (1.1) on [t x , ∞) T .A solution of Eq. (1.1) is said to be oscillatory on [t x , ∞) T in case it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory.Eq. (1.1) is said to be oscillatory in case all its solutions are oscillatory.Our attention is restricted to those solutions of Eq. (1.1) which exist on some half line [t x , ∞) T and satisfy sup{|x(t)| : t ≥ T } > 0 for all T ≥ t x .
Recently, Erbe et al. [7][8][9] studied the oscillatory behavior of third order dynamic equations Hassan [10] and Li et al. [5] considered the oscillation of third order nonlinear delay dynamic equations on time scales (1.4) [5] established some new oscillation criteria for (1.4) that can be applied on any time scale T and the results of [5] are different and complement the results established by [10].
Han et al. [3] considered the oscillation of third order nonlinear delay dynamic equations on time scales x ∆∆ (t) where γ > 0 is a quotient of odd positive integers, p is positive, real-valued and rd-continuous function defined on T, τ : T → T is an rd-continuous function such that τ (t) ≤ t and τ (t) → ∞ as t → ∞, and established some new oscillation criteria for (1.5) which guarantee that every solution of (1.5) oscillates or converges as t → ∞.Yu and Wang [11] studied asymptotic behavior of solutions to more general third-order nonlinear dynamic equations and they showed that if [a i (s)] there exists a positive ∆-differentiable function r on T, for all M > 0 and sufficiently large t 1 , t 2 with t 2 > t 1 , then every solution x of (1.6) is either oscillatory or lim t→∞ x(t) exists (finite).In addition to every solution x of (1.6) is either oscillatory or lim t→∞ x(t) = 0. Clearly, (1.1) is a special case of the above equations.In this paper, we will use a different Riccati transformation with the above papers.The purpose of this paper is to establish some new oscillation criteria for (1.1) which guarantee that every solution x of (1.1) oscillates or converges to zero as t → ∞.Our results are new for third order nonlinear delay dynamic equations and complement the results established in literature.
The paper is organized as follows: In Section 2, we present some lemmas which will be used in the proof of our main results.In Section 3, by developing a Riccati transformation technique, integral averaging technique and inequalities, we give some sufficient conditions which guarantee that every solution of Eq. (1.1) oscillates or converges to zero.In Section 4, we give two examples to illustrate Corollary 3.1 and Theorem 3.2, respectively.

Some preliminary lemmas
In this section, by employing the Riccati transformation technique, we state the main results which guarantee that every solution of Eq. (1.1) oscillates or converges to zero.
It will be convenient to make the following notations: Before stating our main results, we begin with the following lemmas which will play important roles in the proof of the main results.Lemma 2.2 Assume that (1.2) holds.Furthermore, assume that x is an eventually positive solution of (1.1).Then there are only two possible cases for t ≥ t 0 sufficiently large: Proof.Let x be an eventually positive solution of (1.1).Then there exists Then a(t)(x ∆ (t)) α1 is decreasing and there exist constants d and Dividing by b(t) and integrating from t 3 to t, we get 2).Thus, there exist constants c and Dividing by a(t) and integrating the previous inequality from t 4 to t, we have 2), a contradiction with the fact that x(t) > 0. We conclude that (a(t)(x ∆ (t)) α1 ) ∆ > 0 for large t and we get (I) or (II).This completes the proof.
Proof.Let x is an eventually positive solution of (1.1) satisfying Case (I) of Lemma 2.2.Then there exists t 1 ∈ [t 0 , ∞) T and from Eq. (1.1), we have and this leads to (2.4).The proof is complete.

Main results
Now, we are in a position to state and prove the main results which guarantee that every solution of Eq. (1.1) oscillates or converges to zero.
Theorem 3.1 Assume that (1.2) and α 1 α 2 ≥ 1 hold.Furthermore, assume that there exists a positive function and where M is a positive constant, δ is defined as in Lemma 2.3, 1/α1 ∆s.Then every solution of (1.1) is either oscillatory or converges to zero.
Proof.Suppose to the contrary that x is a nonoscillatory solution of (1.1).We may assume without loss of generality that there exists a number t 1 ∈ [t 0 , ∞) T , such that x(t) > 0, x(τ (t)) > 0 and the conclusions of Lemmas 2.1 and 2.2 hold for all t ∈ [t 1 , ∞) T .We only consider the case when x is eventually positive, since the case when x is eventually negative is similar.Since (1.2) holds, in view of Lemma 2.1, there are two possible cases.
Combining (3.7) with (3.8), (3.9) and (3.11), we have Integrating (3.13) from t 2 to t, we obtain for all large t and this leads to a contradiction with (3.1).
Remark 3.2 In [7] and [11], Yu and Wang, Erbe, Peterson and Saker proved that every solution converges to zero if (1.7) holds, respectively.But one can easily see that this result can't be applied if ∞ t0 q(s)∆s < ∞, so our results extend and improve the results in [11].
Remark 3.3 If the assumption (3.2) is not satisfied, we have some sufficient conditions which ensure that every solution of (1.1) oscillates or lim t→∞ x(t) exists (finite).where ξ is defined as in Theorem 3.1.Then every solution of (1.1) is either oscillatory or converges to zero.
Theorem 3.2 Assume that (1.2), (3.2) and α 1 α 2 ≥ 1 hold.Furthermore, assume that there exist m ≥ 1 and a positive function r ∈ C 1 rd ([t 0 , ∞) T , R) such that for sufficiently large t 1 , t 2 with where M, Q and ξ are defined as in Theorem 3.1.Then every solution of (1.1) is either oscillatory or converges to zero.
EJQTDE, 2011 No. 75, p. 8 Proof.Suppose to the contrary that x is a nonoscillatory solution of (1.1).We may assume without loss of generality that there exists a number t 1 ∈ [t 0 , ∞) T , such that x(t) > 0, x(τ (t)) > 0 and the conclusions of Lemmas 2.1 and 2.2 hold for all t ∈ [t 1 , ∞) T .We only consider the case when x is eventually positive, since the case when x is eventually negative is similar.Since (1.2) holds, in view of Lemma 2.1, there are two possible cases.
We define the function ω by (3.3) again and proceeding as in the proof of Theorem 3.1, we have ω(t) > 0 and (3.13).
Multiplying (3.13) by (t − s) m and integrating from t 1 to t, we get Using the integration by parts formula, we obtain where Q(t, s) = ((t − s) m ) ∆s .From (3.16), (3.17) and multiplying 1/t m , we have and noting that m ≥ 1, Q(t, s) ≤ 0 for t ≥ σ(s), we obtain which is a contradiction with (3.15).
Proceeding as in the proof of Theorem 3.1, we get a contradiction with (3.2).This completes the proof.
Remark 3.5 From Theorem 3.2, we can obtain different conditions for oscillation of all solutions of (1.1) with different choices of r.Theorem 3.3 Assume that (1.2), (3.2) and α 1 α 2 ≥ 1 hold.Furthermore, assume that there exist m ≥ 1 and a positive function r ∈ C 1 rd ([t 0 , ∞) T , R) such that for sufficiently large t 1 , t 2 with where M, Q and ξ are defined as in Theorem 3.1 and Then every solution of (1.1) is either oscillatory or converges to zero.
EJQTDE, 2011 No. 75, p. 9 Proof.Suppose to the contrary that x is a nonoscillatory solution of (1.1).We may assume without loss of generality that there exists a number t 1 ∈ [t 0 , ∞) T , such that x(t) > 0, x(τ (t)) > 0 and the conclusions of Lemmas 2.1 and 2.2 hold for all t ∈ [t 1 , ∞) T .We only consider the case when x is eventually positive, since the case when x is eventually negative is similar.Since (1.2) holds, in view of Lemma 2.1, there are two possible cases.
We define the function ω by (3.3) again and proceeding as in the proof of Theorem 3.1, we have ω(t) > 0 and (3.12).
Multiplying (3.12) by (t − s) m and integrating from t 1 to t, we have and this implies that lim sup P 2 (t, s) ∆s ≤ ω(t 1 ), which contradicts (3.18).

Case (II):
x ∆ (t) < 0, (a(t)(x ∆ (t)) α1 ) ∆ > 0, t ∈ [t 1 , ∞) T .The remainder of the proof is similar to that of Theorem 3.1, so we omit the details.This completes the proof.Theorem 3.4 Assume that (1.2), (3.2) and α 1 α 2 ≥ 1 hold.Furthermore, assume that there exist functions H, h ∈ C rd (D, R), where D ≡ {(t, s) : t ≥ s ≥ t 0 } such that H has a nonpositive continuous ∆−partial derivation H ∆s (t, s) with respect to the second variable and satisfies and for sufficiently large where r ∈ C 1 rd ([t 0 , ∞) T , R) is a positive function, M, Q and ξ are defined as in Theorem 3.1 and h − (t, s) = max{0, −h(t, s)}.Then every solution of (1.1) is either oscillatory or converges to zero.
Proof.Suppose to the contrary that x is a nonoscillatory solution of (1.1).We may assume without loss of generality that there exists a number t 1 ∈ [t 0 , ∞) T , such that x(t) > 0, x(τ (t)) > 0 and the conclusions of Lemmas 2.1 and 2.2 hold for all t ∈ [t 1 , ∞) T .We only consider the case when x is eventually positive, since the case when x is eventually negative is similar.Since (1.2) holds, in view of Lemma 2.1, there are two possible cases.EJQTDE, 2011 No. 75, p. 10 We define the function ω by (3.3) again and proceeding as in the proof of Theorem 3.1, we have ω(t) > 0 and (3.12).

Lemma 2 . 1 [ 21 ]
Let a, b ∈ T and τ ∈ C 1 rd ([a, b] T , T) be a strictly increasing function and x