Forced Oscillation of Second-order Nonlinear Dynamic Equations on Time Scales ∗

In this paper, by defining a class of functions, we establish some oscillation criteria for the second order nonlinear dynamic equations with forced term x �� (t) + a(t)f(x(q(t))) = e(t) on a time scale T. Our results unify the oscillation of the second order forced differential equation and the second order forced difference equation. An example is considered to illustrate the main results.


Introduction
The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his PhD thesis in 1988 in order to unify continuous and discrete analysis (see Hilger [19]).The area of dynamic equations on time scales is a new component of applied analysis that describes processes that feature both continuous and discrete elements.Several authors have expounded on various aspects of this new theory, see the survey paper by Agarwal et al. [1] and references cited therein.A book on the subject of time scales, by Bohner and Peterson [7], summarizes and organizes much of the time scale calculus.We refer also to the last book by Bohner and Peterson [8] for advances in dynamic equations on time scales.
We are concerned here with second-order forced nonlinear dynamic equations of the form x ∆∆ (t) + a(t)f (x(q(t))) = e(t) (1.1) on a time scale T, where a(t), q(t) and e(t) are real-valued rd-continuous functions defined on T, the function q(t) also satisfies q : T → T, q(t) → ∞ as t → ∞, and f ∈ C(R, R), xf (x) > 0 whenever x = 0. Agarwal and Grace [4], Ou and Wong [27], Wong [28] studied the n-order forced functional differential equations of the form C ¸akmak and Tiryaki [11] studied (1.1) in the case when T = R.Sun and Saker [26] considered the n-order forced delay difference equations of the form In recent years, there has been much research activity concerning the oscillation of solutions of various second order dynamic equations on time scales, we refer the reader to the articles [2, 3, 5, 6, 9, 10, 12-17, 20, 21, 23-25] and references cited therein.Agarwal et al. [3] investigated the second order forced dynamic equations with mixed nonlinearities [5] considered the oscillations of the forced dynamic equations and Anderson [6] studied the second order dynamic equations Bohner and Tisdell [10] examined oscillation and nonoscillation for (rx ∆ ) ∆ (t) + p(t)x σ (t) = e(t), t ∈ T.
Huang and Feng [20] considered the following second-order forced nonlinear dynamic equations x ∆∆ (t) + p(t)f (x σ (t)) = e(t), t ∈ T, and in [21] the authors studied the oscillation of the forced dynamic equations Oscillatory criteria for the forced dynamic equations where ∞ t0 |r(s)|∆s < ∞ are analyzed in [24].
Philos-type functions [22] are used extensively in the theory of oscillations, for example, Erbe et al. [12], Han et al. [17], Saker et al. [25] established some Philos-type oscillation criteria for the second-order delay dynamic equations on time scales.In [12], the authors utilized the class of functions as follows: Assume H is defined for t 0 ≤ s ≤ σ(t), t, s ∈ [t 0 , ∞) T , H(t, s) ≥ 0, H(σ(t), t) = 0, H ∆s (t, s) ≤ 0 for t ≥ s ≥ t 0 , and for each fixed t, H ∆s (t, s) is delta integrable with respect to s.
As we are interested in oscillatory behavior, we assume throughout this paper that the given 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 of the form In this section, we give some new oscillation criteria for (1.1).Motivated by [22], let us introduce the class of functions H which will be extensively used in the sequel.
It is easy to see that H ∈ H. Therefore, we have the following result.
It is easy to see that H ∈ H. Therefore, we have the following result.

Example
As an application, we consider the following example.