On Nonlinear Forced Impulsive Differential Equations under Canonical and Non-Canonical Conditions

: This study is connected with the nonoscillatory and oscillatory behaviour to the solutions of nonlinear neutral impulsive systems with forcing term which is studied for various ranges of of the neutral coefﬁcient. Furthermore, sufﬁcient conditions are obtained for the existence of positive bounded solutions of the impulsive system. The mentioned example shows the feasibility and efﬁciency of the main results.


Introduction
The study of oscillation of solutions by imposing impulse controls can be found in an extensive variety of real phenomena in Applied Sciences and Engineering problems. Impulsive differential systems arise in bifurcation analysis, circuit theory, population dynamics, biotechnology, loss less transmission in computer network, mathematical economic, chemical technology, etc.
Many researchers spend their attentions to dynamical behaviours of a neutral impulsive differential system (IDS) because it has various applications; an interesting study of second-order impulsive differential systems appears in the theory of impact, as there is a good relation between impact and impulse. The term impulse is also used to refer to a fast-acting force or impact. This type of impulse is often idealized so that the change in momentum produced by the force happens with no change in time. Then, models describing viscoelastic bodies colliding systems with delay and impulses are more appropriate (see [1] and references therein for a review). The models appear in the study of several real-world problems (see, for instance, [2][3][4]). In general, it is well-known that several natural phenomena are driven by impulsive differential equations. Examples of the aforementioned phenomena are related to population dynamics, biological and mechanical systems, pharmacokinetics, biotechnological processes, theoretical physics, chemistry, control theory [5,6] and engineering. Another interesting application is in some vibrational problems [1]. We refer the readers to [7][8][9][10][11] for further details. Many other interesting results concerning nonlinear equations with symmetric kernels with the application of group symmetry have remained beyond the scope of this paper.
Shen et al. [12] considered the IDS of the form: when q, I i ∈ C(R, R) for i ∈ N, and obtained some conditions to ensure the oscillatory and asymptotic behaviour of the solutions of Equation (1). Graef et al. [13] have studied the IDE of the form: where p(ζ) ∈ PC([ζ 0 , ∞), R + ) obtained some results for the oscillation to the solutions of the impulsive differential equations in Equation (2). Shen et al. [14] considered the first-order IDS of the form: and established some new sufficient conditions for oscillation of Equation (3) assuming  (3) with variable delays and extended the results of [14].
Tripathy et al. [16] have studied the oscillation and nonoscillation properties for a class of second-order neutral IDS of the form: with constant delays and coefficients. Some new characterizations related to the oscillatory and the asymptotic behaviour of solutions of a second-order neutral IDS were established in [17], where tripathy and Santra studied the systems of the form: Tripathy et al. [18] have considered the first-order neutral IDS of the form and established some new sufficient conditions for the oscillation of Equation (6) for different values of the neutral coefficient p. Santra et al. [19] obtained some characterizations for the oscillation and the asymptotic properties of the following second-order highly nonlinear IDS: where Tripathy et al. [20] studied the following IDS: where f (ζ) = u(ζ) + p(ζ)u(δ(ζ)) and −1 < p(ζ) ≤ 0 and obtained different conditions for oscillations for different ranges of the neutral coefficient. Finally, we mention the recent work [21] by Marianna et al., where they studied the nonlinear IDS with canonical and non-canonical operators of the form and established new sufficient conditions for the oscillation of solutions of Equation (9) for various ranges of the neutral coefficient p.

Qualitative Behaviour under the Canonical Operator
This section deals with the sufficient conditions for the oscillatory and asymptotic properties of solutions of a nonlinear second-order forced neutral IDS of the form (S) under the canonical operator (H5).
Then each solution of the system (S) is oscillatory.

Qualitative Behaviour under the Noncanonical Operator
In the following, we establish sufficient conditions that guarantee the oscillation and some asymptotic properties of solutions of the IDS (S) under the noncanonical condition (H15).
Proof. The proof of the theorem follows the proof of Theorem 5.

Sufficient Conditions for Nonoscillation
This section deals with the existence of positive solutions to show that the IDS (S) has positive solution. nonincreasing.

Remark 1.
It is not possible to use the Lebesgue's dominated convergence theorem for another intervals of the neutral coefficient except −1 ≤ p(ζ) ≤ 0 as there are different solutions in different ranges. But, one can use Banach's fixed point theorem for another intervals of the neutral coefficient similar to Theorem 8.

Discussion and Example
In this paper, we have seen that (H7)-(H14) and (H16)-(H23) are the new sufficient conditions for oscillatory behaviour of solutions of (S), in which we are depending explicitly on the forcing function. The results of this paper are not only true for (S) but also for its homogeneous counterpart.
Next, we mentioning examples to show feasibility and efficiency of main results.

Example 1. Consider the IDS
where h(θ i ) = Thus, every condition of Theorem 1 is satisfied, and hence, each solution of (S 1 ) is oscillatory by Theorem 1.