Oscillation condition for first order linear dynamic equations on time scales

. In this paper, we deal with the first-order dynamic equations with nonmonotone arguments


Introduction and Background
We consider the delay dynamic equation with several delays which are not necessarily monotone where T is a time scale unbounded above with ζ 0 ∈ T, r i ∈ C rd ([ζ 0 , ∞) T , R + 0 ), ψ i ∈ C rd ([ζ 0 , ∞) T , T) do not have to be monotone for 1 ≤ i ≤ m such that ψ i (ζ) ≤ ζ for all ζ ∈ T, lim ζ→∞ ψ i (ζ) = ∞. (1.2) First of all, we would like to remind some basic concepts about time scales calculus. A function r : T → R is said to be positively regressive (it means that r ∈ R + ) if it is rd-continuous and satisfies 1 + µ(ζ)r(ζ) > 0 for all ζ ∈ T, where µ : T → R + 0 is the graininess function defined by µ(ζ) := σ(ζ) − ζ with the forward jump operator σ : T → T defined with the help of σ := inf{s ∈ T : s > ζ} for ζ ∈ T. If σ(ζ) = ζ or µ(ζ) = 0, a point ζ ∈ T is said to be right-dense, otherwise it is right-scattered.

Ö .ÖCALAN and N. KILIÇ
A function y : T → R is called a solution of (1.1), if y(ζ) is delta differentiable for ζ ∈ T κ and satisfies (1.1) for ζ ∈ T κ . It is called that a solution y of (1.1) has a generalized zero at ζ if y(ζ) = 0 or if µ(ζ) > 0 and y(ζ)y(σ(ζ)) < 0. Let sup T = ∞ and then a nontrivial solution y of (1.1) is called oscillatory on [ζ, ∞) if it has arbitrarily large generalized zeros in [ζ, ∞). Also, we refer to book of Bohner and Peterson [2,3] for more detailed information. For m = 1, we have the following equation which is the form of (1.1) with single delay.
Recently, there has been remarkable interest for the oscillatory solutions of this equation. See  and the references cited therein. Concerning Eq. (1.3) which have monotone arguments, see also Zhang and Deng [20], Bohner [4], Zhang et al. [21], Ş ahiner and Stavroulakis [19], Agarwal and Bohner [1], and Karpuz andÖcalan [9]. As you seen, many articles have been dedicated to the equations which have monotone terms, but a few is related with the more general case of nonmonotone delay terms. Now, we mention the results which contain delay arguments which are not necessarily monotone. Suppose that ψ(ζ) does not have to be monotone and Obviously, ϑ(ζ) is nondecreasing and ψ(ζ) ≤ ϑ(ζ) for all ζ ≥ 0.
Although dynamic equations with several arguments are more comprehensive than dynamic equations with one delay, there are not many studies on this subject. So, in this article, we are interested in studying the oscillatory behavior of first order dynamic equations with several delays on time scale. We present one criterion to check the oscillation of (1.1). Our result is an extension and complement to some results published in the literature.

Main Results
In this section, we introduce a new sufficient condition for the oscillatory solutions of (1.1) when the arguments ψ i (ζ) do not have to be monotone for 1 ≤ i ≤ m and 0 < α ≤ 1 e . The following lemmas will be useful to obtain our main result.
The lemma given below can be easily obtained from [4].
rj (ζ, s) y(s) for all ζ ≥ s, s, ζ ∈ T. (2.1) The result given below can be easily produced by applying a nearly same procedure to [21, Lemma 2.4] when the case ψ i (ζ) do not have to be monotone for 1 ≤ i ≤ m. Therefore, the proof of this lemma is not presented here.
Proof. Assume, for the sake of contradiction, that there exists an eventually positive solution y(ζ) of (1.1). If y(ζ) is an eventually negative solution of (1.1), the proof of the theorem can be done similarly. Then there exists ζ 1 > ζ 0 such that y(ζ), y (ψ i (ζ)) , y (ψ(ζ)) , y (ϑ(ζ)) > 0 for all ζ ≥ ζ 1 and 1 ≤ i ≤ m. So, using (1.1) we obtain which implies that y(ζ) is an eventually nonincreasing function. From this fact and taking into account that and then, we obtain the below expression from Lemma 2.1.
On the other hand, integrating (1.1) from ϑ(ζ) to σ(ζ) and with the help of (2.5), we have Consequently, from (2.6) we obtain and from (2.2) the last inequality turns into which contradicts to (2.3) and this completes the proof. ■
If T = hZ, from Theorem 1.79 [2], we know the formula given below.