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Interval criteria for oscillation of second-order self-adjoint impulsive differential equations

Author: Jitsuro Sugie
Journal: Proc. Amer. Math. Soc. 148 (2020), 1095-1108
MSC (2010): Primary 34A37, 34C10, 34C29
Published electronically: December 6, 2019
MathSciNet review: 4055937
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Abstract: In this paper, oscillation theorems are given for second-order self-adjoint impulsive differential equations. The obtained results extend the well-known Kamenev-type and Philos-type oscillation theorems. A generalized Riccati transformation is used to prove these results. There are two advantages of using the generalized Riccati transformation rather than the standard Riccati transformation. One is that Kamenev-type and Philos-type oscillation theorems cannot be applied to conditionally oscillatory differential equations such as Euler’s equations, but the obtained results can be applied even to such equations. The other advantage is the ability to prove that the impulsive differential equation may become oscillatory even if the total impulse is small. A specific example is included to demonstrate the merits of the results obtained.

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Additional Information

Jitsuro Sugie
Affiliation: Department of Mathematics, Shimane University, Matsue 690-8504, Japan
MR Author ID: 168705

Keywords: Oscillation of solutions, impulse, generalized Riccati transformation, integral averaging technique
Received by editor(s): March 22, 2019
Received by editor(s) in revised form: April 28, 2019
Published electronically: December 6, 2019
Additional Notes: This work was supported in part by JSPS KAKENHI Grants-in-Aid for Scientific Research (C)) Grant Number JP17K05327.
Communicated by: Wenxian Shen
Article copyright: © Copyright 2019 American Mathematical Society