First-order impulsive differential systems: sufficient and necessary conditions for oscillatory or asymptotic behavior

In this paper, we study the oscillatory and asymptotic behavior of a class of first-order neutral delay impulsive differential systems and establish some new sufficient conditions for oscillation and sufficient and necessary conditions for the asymptotic behavior of the same impulsive differential system. To prove the necessary part of the theorem for asymptotic behavior, we use the Banach fixed point theorem and the Knaster–Tarski fixed point theorem. In the conclusion section, we mention the future scope of this study. Finally, two examples are provided to show the defectiveness and feasibility of the main results.

In [10], Shen and Wang considered impulsive differential systems of the following form: where b ∈ C(R, R) and I i ∈ C(R, R) for i ∈ N, and established some sufficient conditions that ensure the oscillatory and asymptotic behavior of (1).
In [12], the authors established some new oscillation criteria for first order impulsive neutral delay differential systems of the form under the assumptions that b(ξ ) ∈ PC([ξ 0 , ∞), R + ) and b i ≤ I i (υ) υ ≤ 1. Karpuz et al. in [13] extended the results contained in [12] by taking the nonhomogeneous counterpart of system (3) with variable delays.
In [15], in particular, the authors were interested in oscillating systems that, after a perturbation by instantaneous change of state, remain oscillating.
We also mention the paper [17] in which Santra and Dix, using the Lebesgue dominated convergence theorem, obtained sufficient and necessary conditions for the oscillation of the following second-order neutral differential equations with impulses: where In line with the contents of [17], Tripathy and Santra in [18] examined oscillation and non-oscillation properties for the solutions of the following forced nonlinear neutral impulsive differential system: for different values of b(ξ ) and established sufficient conditions for the existence of positive bounded solutions of system (8).
Finally we mention the recent work [19] in which Tripathy and Santra studied the characterizations for the oscillation of second-order neutral delay impulsive differential system where For further details on neutral impulsive differential equations and for recent results related to the oscillation theory for delay differential equations, we refer the reader to the papers  and to the references therein. In particular, the study of oscillation of halflinear/Emden-Fowler (neutral) differential equations with deviating arguments (delayed or advanced arguments or mixed arguments) has numerous applications in physics and engineering (e.g., half-linear/Emden-Fowler differential equations arise in a variety of real world problems such as in the study of p-Laplace equations, chemotaxis models, and so forth); see, e.g., the papers [4, 44-47, 49, 50, 52, 53] for more details. In particular, by using different methods, the following papers were concerned with the oscillation of various classes of half-linear/Emden-Fowler differential equations and half-linear/Emden-Fowler differential equations with different neutral coefficients (e.g., the paper [43] was concerned with neutral differential equations assuming that 0 ≤ b(ξ ) < 1 and b(ξ ) > 1; in [44] the authors studied neutral differential equations assuming that 0 ≤ b(ξ ) < 1; in [46], the authors considered neutral differential equations assuming that b(ξ ) is nonpositive; in [47,51] the author considered neutral differential equations in the case where b(ξ ) > 1; the paper [50] was concerned with neutral differential equations assuming that 0 ≤ b(ξ ) ≤ q 0 < ∞ and b(ξ ) > 1; in [52] the authors considered neutral differential equations in the case where 0 ≤ b(ξ ) ≤ q 0 < ∞; in [53] the author studied neutral differential equations in the case when 0 ≤ b(ξ ) = b 0 = 1; whereas the paper [49] was concerned with differential equations with a nonlinear neutral term assuming that 0 ≤ b(ξ ) ≤ a < 1), which is the same research topic as that of this paper.
Motivated by the aforementioned findings, in this paper we prove sufficient and necessary conditions for oscillatory or asymptotic behavior of solutions to a first-order nonlinear impulsive differential system in the form is the difference operator defined by (e) there exists F ∈ C(R, R) such that f (ξ ) = F (ξ ) and g(α i ) = F(α i ), i ∈ N. Next, we are listing all the assumptions/conditions which we need to study the oscillation and non-oscillation properties of the solution of system (E).

Sufficient conditions for oscillation
In this section, we establish sufficient conditions for the oscillation of the impulsive system (E).
Proof To prove by contradiction, we follow the proof of Theorem 2.1 to get that H(ξ ) is monotonic on [ξ 2 , ∞). So, we have the following two possible cases. (12) and (13) are reduced to for ξ ≥ ξ 4 . Next, we are going to prove -∞ < lim ξ →∞ H(ξ ) < 0. If not, letting (12) and (13) can be viewed as Integrating (18) from ξ 3 to +∞, we have which is a contradiction to (A7). The case υ(ξ ) < 0 for ξ ≥ ξ 0 is similar. Hence the details are omitted. Thus, the theorem is proved.

Sufficient and necessary conditions for oscillation
In this section, we are going to present the sufficient and necessary condition for oscillatory or asymptotic behavior of system (E).

Let us define
Clearly, S is a convex and closed subspace of X. Let : S → S be an operator defined by For every ξ ∈ S, imply that ( υ) ∈ S. Again, for υ 1 , υ 2 ∈ S, that is, is a contraction mapping with the contraction (a 2 + 1-a 2 5 ) = 1+4a 2 5 < 1. Note that is a contraction on S and S is complete. Then, by using the Banach fixed point theorem, ξ has a unique fixed point on [ 1-a 2 10 , 1]. Hence υ = υ and is a non-oscillatory solution of system (E) on [ 1-a 2 10 , 1] such that lim ξ →∞ υ(ξ ) = 0. Thus, the theorem is proved. Proof The proof of the sufficient part is the same as in the proof of Theorem 3.1.
To prove necessity, we assume that (26) holds. So, ξ 1 > 0 we have where L = max{L 1 , L 2 } and L 1 is the Lipschitz constant of G on [c, d], where L 2 = G(d) such that a = 2c(a 3 2a 4 )a 4 (a 3 + a 3 2 -2) Also, we can find ξ 2 > 0 such that |F(ξ ) -M| < 1 2 (a 3 -1) for ξ ≥ ξ 2 > ξ 1 . Next, we define a Banach space X as in the proof of Theorem 3.1 with respect to the sup norm Clearly, S is a convex and closed subspace of X. Let : S → S be an operator defined by For every υ ∈ S, implies that ∈ S. For υ 1 , υ 2 ∈ S, that is, is a contraction mapping with the contraction ( 1 a 3 + a 3 -1 2a 3 ) < 1. Hence, by the Banach fixed point theorem, has a unique fixed point on [a, b] which is a non-oscillatory (specially positive) solution of system (E).
This completes the proof of the theorem. Proof The proof is totally the same as in the proof of Theorem 3.2, but, for the necessary part, we provide the following settings: This completes the proof of the theorem.

Conclusion
In [20], the author studied the oscillatory behavior of solutions of the impulsive system under the sufficient condition Because of Theorem 3.1 [20], (27) could be a sufficient and necessary condition for the oscillatory and asymptotic behavior of solutions of system (E1) for different ranges of the neutral coefficient b(ξ ). We guess that (27) could be a sufficient and necessary condition for the oscillation of a non-homogeneous counterpart of (E). In this work, we have obtained sufficient conditions for the oscillation of (E), which is presented in Sect. 2, and in Sect. 3 we have established sufficient and necessary conditions for the oscillatory or asymptotic behavior of (E). It would be of interest to examine the oscillation of (E) with a different neutral coefficient; see, e.g., the papers [43,46,47,[50][51][52][53] for more details. Furthermore, it is also interesting to analyze the oscillation of (E) with a nonlinear neutral term; see, e.g., the paper [49] for more details. Remark 4.2 Lemma 3.1 does not hold when b(ξ ) ≡ 1 for all ξ (see, e.g., [54]), and the present study does not allow us when b(ξ ) ≡ -1 for all ξ . Thus, in this paper, we have obtained necessary and sufficient conditions for the oscillatory or asymptotic behavior of (E) except b(ξ ) = ±1 for all ξ . Hence, it is clear that a different method is necessary to study the oscillatory or asymptotic behavior of (E) when b(ξ ) = ±1. However, we have established sufficient conditions for b(ξ ) = ±1 in Sect. 2.

Examples
In this section, we provide two examples to validate our main results.