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The Least Possible Impulse for Oscillating All Nontrivial Solutions of Second-Order Nonoscillatory Differential Equations

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Abstract

Nonoscillatory solutions of second-order linear differential equations with a parameter may become oscillatory with variations in the parameter. Such equations are termed as conditionally oscillatory equations and are classified according to the result established by Einar Hille, wherein Euler-type differential equations are used as scale marks for classification. Moreover, nonoscillatory solutions can be converted into oscillatory solutions by introducing impulse. Thus, this research determines the least possible amount of impulse required to oscillate all nonoscillatory solutions for each classified group of conditionally oscillatory equations without altering the parameter. Presumably, nonoscillatory solutions can be expected to oscillate when a large amount of impulse is added. Therefore, the focus of this paper is to calculate the least possible amount of impulse required for the stated change. A result derived from a Philos-type oscillation theorem for impulsive differential equations is used to establish the proofs.

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Acknowledgements

The author would like to thank Editage (www.editage.com) for English language editing.

Funding

This work was supported in part by a JSPS KAKENHI Grant-in-Aid for Scientific Research (C) [Ggrant Number JP20K03701].

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  1. Department of Mathematics, Shimane University, 1060 Nishikawatsu-cho, Matsue, Shimane, 690-8504, Japan

    Jitsuro Sugie

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  1. Jitsuro Sugie
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Sugie, J. The Least Possible Impulse for Oscillating All Nontrivial Solutions of Second-Order Nonoscillatory Differential Equations. Qual. Theory Dyn. Syst. 21, 43 (2022). https://doi.org/10.1007/s12346-022-00571-4

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