Forced oscillation of second-order superlinear dynamic equations on time scales

In this paper, by constructing a class of Philos type functions on time scales, we investigate the oscillation of the following second-order forced nonlinear dynamic equation x �� (t) − p(t)j x(q(t))j � 1 x(q(t)) = e(t); t 2 T where T is a time scale, p; e : T ! R are right dense continuous functions with p > 0, � > 1 is a constant, and q(t) = t or q(t) = � (t). Our results not only unify the oscillation of second-order forced differential equations and their discrete analogues, but also complement several results in the literature.


Introduction
Following Hilger's landmark paper [1], a rapidly expanding body of literature has sought to unify, extend and generalize ideas from discrete calculus, quantum calculus and continuous calculus to arbitrary time-scale calculus, where a time scale is an arbitrary closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations.Many other interesting time scales exist, e.g., T = q N 0 = {q t : t ∈ N 0 } for q > 1 (which has important applications in quantum theory), T = hN with h > 0, T = N 2 and T = T n the space of the harmonic numbers.For an introduction to time scale calculus and dynamic equations, we refer to the seminal books by Bohner and Peterson [2,3].
A solution of Eq. ( 1) is a nontrivial real function x : and x satisfies Eq. (1) on T. A function x is an oscillatory solution of Eq. ( 1) if and only if x is a solution of Eq. ( 1) that is neither eventually positive nor eventually negative.Eq. ( 1) is oscillatory if and only if every solution of Eq. ( 1) is oscillatory.Some equations related to Eq. ( 1) have been extensively studied by many authors in [20][21][22][23][24][25][26][27].For the oscillation of the second-order forced dynamic Eq. ( 1), the oscillation results in [6] can be applied to Eq. ( 1) with q(t) = σ(t) and oscillatory potentials.Following the idea in [27], the authors established several oscillation criteria for Eq. ( 1) with p(t) > 0 and q(t) = σ(t) in [18] and [19], while the case of q(t) = t remains unstudied.
The main purpose of this paper is to further study the oscillation of Eq. ( 1) in the superlinear case when q(t) = σ(t) and q(t) = t, respectively.We will show that the results in [18] and [19] seem to be invalid when the time scale T only contains isolated points.We also extend the results to the case of q(t) = t.Based on the usual Philos type functions for differential equations, we first construct a class of explicit functions on time scales for Eq.(1).Then, several oscillation criteria for Eq.(1) are established in both the case q(t) = σ(t) and the case q(t) = t, which complement those results in [18] and [19].

Time scale essentials
The definitions below merely serve as a preliminary introduction to the time-scale calculus; they can be found in the context of a much more robust treatment than is allowed here in the text [2] and the references therein.Definition 2.1 Define the forward (backward) jump operator σ(t) at t for t < sup T (respectively ρ(t) at t for t > inf T) by The graininess functions are given by µ(t) = σ(t) − t and v(t) = t − ρ(t).
Throughout this paper, the assumption is made that T is unbounded above and has the topology that it inherits from the standard topology on the real numbers R. Also assume EJQTDE, 2011 No. 44, p. 2 throughout that a < b are points in T. The jump operators σ and ρ allow the classification of points in a time scale in the following way: If σ(t) > t the point t is right-scattered, while if ρ(t) < t then t is left-scattered.Points that are right-scattered and left-scattered at the same time are called isolated.If t < sup T and σ(t) = (t) the point t is right-dense; if t > inf T and ρ(t) = t then t is left-dense.Points that are right-dense and left-dense at the same time are called dense.The composition Definition 2.3 Fix t ∈ T and let y : T → R. Define y ∆ (t) to be the number (if it exists) with the property that given ǫ > 0 there is a neighborhood U of t such that, for all Call y ∆ (t) the (delta) derivative of y at t.The following theorem is due to Hilger [1].
(5) If f and g are differentiable at t, then f g is differentiable at t with EJQTDE, 2011 No. 44, p. 3

Main results
We first construct Philos type functions on time scales for Eq.(1).In [18] and [19] the authors only sketchily defined Philos type functions on time scales for Eq. ( 1), while they did not answer how to construct these functions explicitly. Let Recall to introduce a usual Philos type function class X in [26] and [27].The function Just as shown in the sequel, the results in [18] and [19] seem to be invalid when the time scale T only contains isolated points.Therefore, we are here concerned with the time scale T which only contains isolated points.Now, based on any functions H 1 , H 2 ∈ X , we define the following Philos type function class on time scales for Eq. ( 1) where Then, H(t, σ(s)) ≥ 0 for t 0 ≤ s ≤ t and H(t, σ(s)) = 0 only holds at s = t.Straightforward computation yields and It is not difficult to verify Before giving the main results of this paper, we first recall to introduce Theorem 2.2 in [18] as followings: Theorem A [18].Assume that q(t) = σ(t where then Eq. ( 1) with q(t) = σ(t) is oscillatory.
We show that Theorem A seems to be invalid for the case when the time scale T only contains isolated points.In fact, in the proof of Theorem 2.2 in [18], the authors used a basic inequality to estimate Note that H(σ(t), σ(t)) = 0 and t is an isolated point, we do not use the inequality (5) to get that Based on the definition of H(t, s), we can only conclude that Therefore, the term remains unestimated.
To complement those results in [18] and [19], we here focus on the oscillation of Eq.
(1) on time scales which only contain isolated points.
Taking lim sup on both sides of the above inequality as t → ∞ and using conditions ( 6)-( 8), we get a desired contradiction.This completes the proof of Theorem 3.1.
where Proof.Let x(t) be a nonoscillatory solution of Eq. (1).Say x(t) > 0 for t ≥ t 0 and t ∈ T. Multiplying Eq. ( 1) by H(t, σ(s)) and integrating from t 0 to σ(t) by the integration by parts formula, we get This together with ( 13)-( 15) yield a contradiction.The proof of Theorem 3.2 is complete.
For the special case T = Z, we have the following oscillation results:  To illustrate the usefulness of the results, we state the corresponding theorems in the above for the special case T = Z.It is not difficult to provide similar results for other specific time scales of interest.On the other hand, all the results obtained in this paper are restricted to those solutions satisfying |x(t)| = O(φ(t)).At present, it seems difficult to obtain sufficient conditions for the oscillation of all solutions of Eq. (1) with p(t) > 0 and λ > 1 when the time scale T only contains isolated points.This problem is left for future study.
is continuous at each right-dense point and if there exists a finite left limit in all left-dense points.Every right-dense continuous function has a delta antiderivative [2, Theorem 1.74].This implies that the delta definite integral of any right-dense continuous function exists.Likewise every left-dense continuous function f on the time scale, denoted f ∈ C ld (T, R), has a nabla antiderivative[2, Theorem 8.45]