2023 Fiscal Year Final Research Report
Innovation in Qualitative Analysis of Nonlinear Differential Equations: Entering a New Phase of Nonlinear Oscillation Theory
| Project/Area Number |
16K05238
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Mathematical analysis
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| Research Institution | Osaka Metropolitan University (2022-2023)
Osaka Prefecture University (2018-2021)
Kumamoto University (2016-2017) |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2024-03-31
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| Keywords | 微分方程式の振動理論 / 非線形微分方程式の振動理論 / 正則変動関数論 |
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| Outline of Final Research Achievements |
Nonlinear Sturm-Liouville differential operators (both non-neutral and neutral types) and their associated systems of differential equations: (a) Characterization of oscillatory behavior in nonlinear differential equations, (b) Effect of adding a non-purturbative term to a differential equations of non-oscillatory types, (c) Utilization of Regular Variation Theory for analyzing non-oscillatory solutions of various differential equations. Regular variation functions are a general term for a family of classes of functions classified according to certain criteria. (d) Relationship between non-oscillatory solutions of nonlinear differential equations and nonlinear Riccati equations, which provide useful informations such as the existence of non-oscillatory solutions. Explored the connection between non-oscillatory solutions of nonlinear differential equations and solutions of nonlinear Riccati equations.
Provided valuable insights into the existence of non-oscillatory solutions.
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| Free Research Field |
数物系科学
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| Academic Significance and Societal Importance of the Research Achievements |
本研究課題の学術的意義： 非線形微分方程式の振動理論は, 自然現象や社会現象を記述する数理モデル（非線形微分方程式）の解の全体構造を解明することを目標に世界各国で長年研究されてきた分野である. その振動理論の進展に大きく寄与することができたと思える. また, 本来, 非線形微分方程式に対する解は具体的に表現することができないため, 上述のような振動理論を駆使して解の性質を捉えることが最善の手法になっている.
本研究の社会的意義： 歴史のある振動理論を活用することによって, 多種多様の微分方程式の解の特徴を入手することに成功することができ, 科学及び文明の進展に大きく貢献したと思われる.
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