On the Oscillation of Impulsive Neutral First-order Differential Equations with Variable Arguments

Euat Tallah, Samah Saad; Elsheikh, M. M. A.; Ismail, G. A. F.

doi:10.21608/jsrs.2022.275766

On the Oscillation of Impulsive Neutral First-order Differential Equations with Variable Arguments

Document Type : Original Article

Authors

¹ Department of Mathematics, University College for women, Ain shams university, Cairo, Egypt.

² Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El-Koom, Egypt.

³ Department of Mathematics, Faculty of Women for Arts, Science and Education, Ain Shams University, Cairo, Egypt.

10.21608/jsrs.2022.275766

Abstract

Throughout the article, we study the oscillation of a general class of first-order neutral differential equations in presence of variable delays under the effect of impulses.
Due to its importance in applications, there are many papers concerning with the property of oscillation and non-oscillation of neutral delay differential equations. Although, a lot of works are concerning with the oscillation of neutral delay differential equations without impulse or impulsive neutral with constant delays, however few papers dealt with the impulsive neutral and those with variable delays. In this paper, we establish sufficient conditions of certain neutral equations with variable delay arguments. New oscillation criteria are deduced. Our results are based on using equivalence transformation and two useful lemmas to prove the obtained criteria.
The results of this paper improve those of [20] by adding several non-linear delay functions to the equations instead of having one delay term. Where it is assumed that the two variable delays satisfying a Lipschitz condition.
Moreover we discuss more general non-linear delay functions comparing with those used in [14]. Our results improve and extend some recent results in the literature. An illustrative example is given.

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