Second-Order Impulsive Delay Differential Systems: Necessary and Sufficient Conditions for Oscillatory or Asymptotic Behavior

In this work, we aimed to obtain sufficient and necessary conditions for the oscillatory or asymptotic behavior of an impulsive differential system. It is easy to notice that most works that study the oscillation are concerned only with sufficient conditions and without impulses, so our results extend and complement previous results in the literature. Further, we provide two examples to illustrate the main results.


Introduction
Nowadays, impulsive differential equations are attracting a lot of attention. They appear in the study of several real world problems (see, for instance, [1][2][3]). In general, it is well-known that several natural phenomena are driven by differential equations, but the descriptions of some real world problems subjected to sudden changes in their states have become very interesting from a mathematical point of view because they should be described while considering systems of differential equations with impulses. Examples of the aforementioned phenomena are related to mechanical systems, biological systems, population dynamics, pharmacokinetics, theoretical physics, biotechnology processes, chemistry, engineering and control theory.
The literature related to impulsive differential equations is very vast. Here we mention some recent developments in this field.
In [4], Shen and Wang considered impulsive differential equations with the following form: where r, I j ∈ C(R, R) (that is, r, I j are continuous in (R, R)) for j ∈ N, and established some sufficient conditions for the oscillatory and the asymptotic behavior of the solutions of the problem (1).
In [8], Tripathy and Santra considered the the impulsive system and studied oscillation and non-oscillation properties for (4). In an another paper, Tripathy and Santra studied the following impulsive systems: In [9], in particular, the authors are interested in oscillating systems that, after a perturbation by instantaneous change of state, remain oscillating.
Motivated by the aforementioned findings, in this paper we prove necessary and sufficient conditions for the oscillatory or asymptotic behavior of solutions to a secondorder non-linear impulsive differential system of the form and the functions g, b, c,c, a, ϑ, σ are continuous such that , (in general C k means the function has k derivatives and they are all continuous functions) (A5) The sequence {χ j } satisfying 0 < χ 1 < χ 2 < · · · < χ j < · · · → ∞ as j → ∞ are fixed moments of impulsive effects; (A6) ν is the quotient of two positive odd integers. In particular, the assumption of ν can be replaced by ν > 0, by using |u| ν sgn(u) instead of u ν , but the notation will be much longer.
From (12), we have a(ξ) w (ξ) µ is non-increasing, including impulses for ξ ≥ ξ * . By contradiction we assume that a(ξ) w (ξ) µ ≤ 0 at a certain time ξ ≥ ξ * . Using that c is not identically zero on any interval [b, ∞), and that g(ξ) > 0 for ξ > 0, by (12), there exist Then Integrating from ξ 1 to ξ, we have Using (A4), we arrive at lim ξ→∞ w(ξ) = −∞. Since b is bounded and w is unbounded, u can not be bounded. This allows the existence of a sequence Then, there is ξ 1 ≥ ξ * such that only one of the following two cases happens.
Proof. Assume the contrary and suppose that (E1) has a non-oscillatory solution u which is positive and does not converge to zero. Then, case 1 in Lemma 1 leads to lim ξ→∞ u(ξ) = 0, which a contradiction. Case 2 of Lemma 1 also leads to a contradiction. In case 2 there exists ξ 1 such that Now, we see that w is left continuous at χ j , It is clear that Λ(ξ) > 0 for ξ ≥ ξ 1 . Computing the derivative, To estimate the discontinuities of Λ 1−α/µ we use a first order Taylor polynomial for the function h(u) = u 1−α/µ , with 0 < α < µ, about u = e: .
. Integrating (18) from ξ 2 to ξ, and using that Λ > 0, we have Since w ≤ u, by (A3), (15), (10), and (17), we have Since α/µ > 0 and ϑ(s) < s, we have Going back to (19), we have which contradicts (16). This completes the proof of sufficient part of the theorem when the solution is a eventually positive. For an eventually negative solution u, we will define a new variable v = −u and g(ξ) = −g(ξ). Then v is an eventually positive solution of (E1) with f instead of g. We find that f satisfies (A3) and (15). Therefore, the above method can be applied to the v solution.
Next, by a contrapositive argument, we show the necessity part-that is, if (16) is not true then there is a non-oscillatory solution. Let (16) be untrue for some δ > 0; then for each > 0 there exists for all s ≥ ξ 1 . In particular we use a positive such that Note that ξ 1 depends on δ. We define Then we can define an operator Φ on S as follows: Now we are going to show that u is a fixed point of Φ in S, that is, Φu = u; u is an eventually positive solution: of (E1).
For u ∈ S, we have 0 ≤ 1/µ A(ξ) ≤ u(ξ). By (A3), we have 0 ≤ g u(ϑ(s)) and by (A2) we have For u in S, by (A2) and (A3), we have g u(ϑ(ζ)) ≤ g δa(ϑ(ζ)) . Then by (22) and (23), Therefore, Φ maps S to S. In the next section, we search a fixed point for Φ in S. Let us define a recurrence relation Note that for each fixed v, we have v 1 (ξ) ≥ v 0 (ξ). Using the mathematical induction and the fact that g is non-decreasing, one can prove v n+1 (ξ) ≥ v n (ξ). Therefore, u is a fixed point of Φ in S; that is, Φu = u by using the lebesgue dominated convergence theorem. Thus, we have a eventually positive solution. This completes the proof. In the next theorem, we assume that ϑ 0 is a differentiable function, such that Additionally, we assume that there exists a constant α, with µ < α, such that g(u) u α is non-decreasing for 0 < u where α is a ratio of two positive odd integers. For example, g(u) = |u| β sgn(u), with α < β, satisfies this condition.
For eventually negative solutions, we will use the same variables that were defined in Theorem 1, and follow the same method used in Theorem 1.
For the necessary part, we suppose that (26) does not suffice, and obtain an eventually positive solution that does not converge to zero. Let (26) not hold; then for each > 0 there exists ξ 1 ≥ ξ 0 such that In particular we use = g(2/(1 − b 0 ) −1/µ > 0. Let us consider the set of continuous functions. Then we define the operator Note that if u is continuous, Φu is also continuous at ξ = ξ 1 . This follows by taking the right and left limits in the three possible cases in the the definition of Φ. Additionally, if Φu = u, then u is solution of (E1).
The rest of necessary part follows from Theorem 1.
The next theorem does not require neither (15) nor (25), but considers only bounded solutions. Proof. We prove sufficiency by contradiction. Suppose u is an eventually positive solution that does not converge to zero. Then we proceed as in Lemma 1 up to Equation (13). u and b are bounded, so w is bounded. Then the left-hand side of (13) is bounded, and the right-hand side approaches −∞ as ξ → ∞. This contradiction implies that w (ξ) > 0 for ξ ≥ ξ 1 . As in Lemma 1, we find two possible cases.
Case 1: w(ξ) < 0 for all ξ ≥ ξ 1 . This leads to a contradiction. As in case 1 of Lemma 1, we have lim ξ→∞ u(ξ) = 0, which contradicts the assumption that u does not converge to zero.