Improvements on the Leighton-type oscillation criteria for impulsive differential equations and extension to non-canonical case

IF 1.8 3区数学 Q1 MATHEMATICS, APPLIED Mathematical Methods in the Applied Sciences Pub Date : 2024-11-11 DOI:10.1002/mma.10605

A. Zafer, S. Doğru Akgöl

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Abstract

We investigate the oscillatory behavior of solutions of second-order linear differential equations with impulses. These equations can be categorized into two types: canonical and non-canonical. This study examines the canonical and non-canonical impulsive differential equations in connection with the famous Leighton-type oscillation theorem. The well-known theorem states that every solution of

This pioneering work of Leighton has received considerable attention since its inception, and hence, it has been extended to various types of equations, including delay differential equations, dynamic equations, and impulsive differential equations. There are also studies generalizing the Leighton oscillation theorem when the condition $(*)$ fails, that is, when the equation is of non-canonical type. Our work focuses on refining the Leighton oscillation theorem to treat the canonical and non-canonical cases. By doing so, we correct, supplement, and enhance the current literature on the oscillation theory of differential equations with impulses. Examples are also given to illustrate the significance of the obtained results.

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脉冲微分方程的leighton型振荡判据的改进及其在非正则情况下的推广

研究了一类二阶脉冲线性微分方程解的振动性。这些方程可以分为两类：正则和非正则。本文结合著名的leighton型振荡定理，研究了典型和非典型脉冲微分方程。这个著名的定理表明，Leighton的这项开创性工作的每一个解从一开始就受到了相当大的关注，因此，它被推广到各种类型的方程，包括延迟微分方程、动态方程和脉冲微分方程。当条件（∗）$$ \left(\ast \right) $$不成立时，即当方程是非正则型时，也有研究推广雷顿振荡定理。我们的工作重点是改进雷顿振荡定理来处理正则和非正则情况。通过这样做，我们纠正，补充，并加强了目前关于脉冲微分方程振荡理论的文献。文中还举例说明了所得结果的意义。

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来源期刊

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.

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