Abstract
In this paper, some necessary and sufficient conditions are obtained to ensure the oscillatory of all solutions of the first order impulsive neutral differential equations. Also, some results in the references have been improved and generalized. New lemmas are established to demonstrate the oscillation property. Special impulsive conditions associated with neutral differential equation are submitted. Some examples are given to illustrate the obtained results.
Highlights
The oscillatory theory of impulsive delay differential equations is appearing as an important field of investigation, because it is much richer than the theory of delay differential equations without impulses effects
The importance of the need to study differential equations with impulsive is due to the fact that these equations are more comprehensive in their use of mathematical modeling where gaps in the model can be addressed by limiting these gaps in specific points called the points of impulses in many real processes and phenomena studied in control theory, biology, mechanics, medicine, electronic, economic, etc
There are a lot of applications of impulsive differential equations in neural networks (1-4), n control theory (5), in biology (6) and economics (7)
Summary
The oscillatory theory of impulsive delay differential equations is appearing as an important field of investigation, because it is much richer than the theory of delay differential equations without impulses effects. The main purpose of this paper is to study one of the important properties of the solution for the impulsive neutral differential equations with positive and negative coefficients, it is the oscillation property. The sufficient conditions to guarantee the oscillation of all solutions for the impulsive neutral differential equation with positive and negative coefficients have been obtained. A function x(t) ∈ ∁([t0, ∞), R) is said to be eventually positive (negative), if there exist t1 ≥ t0 such that x(t) > 0 (x(t) < 0) for all t ≥ t1. Some Basic Lemmas: The following lemmas will be useful to prove the main results: Lemma 1: (13) Suppose that g, h: [t0, ∞) → R are continuous functions, g(t) ≥ 0 eventually, h(t) ≥ t and h′(t) ≥ 0 for t ≥ t0. In view of condition (7) and Lemma 1, the last inequality cannot has eventually positive solution, which is a contradiction.
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