2004 Fiscal Year Final Research Report Summary
Asymptotic analysis of nonlinear ordinary differential equations and its applications
| Project/Area Number |
14540177
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | HIROSHIMA UNIVERSITY |
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Principal Investigator |
USAMI Hiroyuki Hiroshima University, Faculty of Integrated Arts and Sciences, Associate Professor, 総合科学部, 助教授 (90192509)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHIDA Kiyoshi Hiroshima University, Faculty of Integrated Arts and Sciences, Professor, 総合科学部, 教授 (80033893)
MIZUTA Yoshihiro Hiroshima University, Faculty of Integrated Arts and Sciences, Professor, 総合科学部, 教授 (00093815)
SHIBATA Tetsutaro Hiroshima University, Graduate School of Engineering, Professor, 大学院・工学研究科, 教授 (90216010)
NAITO Manabu Ehime University, Faculty of Science, Professor, 理学部, 教授 (00106791)
NAITO Yuki Kobe University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (10231458)
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| Project Period (FY) |
2002 – 2004
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| Keywords | oscillation / elliptic partial differential equation / quasilinear / eigenvalue problem / variational problem / polyharmonic function / parabolic partial differential equation / ordinary differential equations |
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| Research Abstract |
1.We consider second-order quasilinear ordinary differential equations which can be regarded as generalization of the Emden-Fowler equation. We determine asymptotic forms of every positive solutions. We also establish uniqueness of several types of positive solutions.
2.We consider elliptic systems of Emden-Fowler type. We establish sufficient conditions for the oscillation of all solutions to the system. When the coefficient functions behave like positive constant multiples of |x|^a, our conditions are best possible in some sense.
3.Until now oscillatory properties of second order quasilinear elliptic equations have been studied under several additional assumptions imposed on the nonlinear terms. However, we can establish effective oscillation criteria without doing so. In particular, for autonomous equations we can establish necessary and sufficient conditions for the oscillation of all solutions.
4.Eigenvalue problems are studied for second-order semilnear ordinary differential equations, as well as partial differential equations of elliptic type. Asymptotic forms are obtained for variational eigenfunctions and eigenvalues.
5.Asymptotic behavior of evetually positive solutions of n-th order quasilinear ordinary differential equations is studied. We establish necessary and sufficient conditions for the existence of eventually positive solutions with specified asymptotic behavior near the infinity.
6.Fourth-order quasilinear ordinary differential equations are studied. We can establish necessary and/or sufficient conditions for such equations to have no eventually positive solutions.
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