| Project/Area Number |
13640178
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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 | EHIME UNIVERSITY |
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Principal Investigator |
NAITO Manabu Ehime University, Faculty of Science, Professor, 理学部, 教授 (00106791)
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| Co-Investigator(Kenkyū-buntansha) |
USAMI Hiroyuki Hiroshima University, Faculty of Integrated Arts and Sciences, Associate Professor, 総合科学部, 助教授 (90192509)
HASHIMOTO Takahiro Ehime University, Faculty of Science, Instructors, 理学部, 助手 (60291499)
SAKAGUCHI Sigeru Ehime University, Faculty of Science, Professor, 理学部, 教授 (50215620)
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| Project Period (FY) |
2001 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| Budget Amount *help |
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2002: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2001: ¥1,400,000 (Direct Cost: ¥1,400,000)
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| Keywords | nonlinear differential equation / higher-order differential equation / nonoscillatory solution / zero / eigenvalue problem / Liouville's theorem / poly-harmonic operator / 正値解 |
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| Research Abstract |
The aim of this research is to study the number of zeros and the distribution of zeros of solutions of higher-order ordinary differential equations with Emden-Fowler type nonlinearity, and to discuss the oscillatory properties of solutions of higher-order elliptic differential equations on the base of the results for ordinary differential equations.
The new results and knowledge obtained in the two years are as follows :
1. For a singular eigenvalue problem to the linear and sublinear higher-order ordinary differential equations on an infinite interval [a, +∞), it is shown that there is a countable sequence of eigenvalues and that the n-th eigenfunction has exactly n zeros.
2. For a regular eigenvalue problem to the sublinear higher-order ordinary differential equations on a finite interval, it is shown that there is a countable sequence of eigenvalues and that the n-th eigenfunction has exactly n zeros.
3. For the second-order half-linear ordinary differential equations, it is shown that the number of zeros of specific nonoscillatory solutions changes one by one as a parameter varies.
4. For the second-order half-linear ordinary differential equations, a generalization and an analogue of the Sturm-Liouville linear regular eigenvalue problem are obtained.
5. For the four-dimensional Emden-Fowler differential systems and the fourth-order quasilinear differential equations of Emden-Fowler type, a necessary and sufficinet condition for the existence of nonoscillatory solutions with specific asymptotic properties as t →∞ is established, and a sufficient condition for oscillation of all solutions is also obtained.
6. For the two-dimensional semilinear elliptic differential systems of the Laplace type, an analogue of Liouville's theorem is established.
7. For the fourth-order nonlinear elliptic differential equations including the poly-harmonic operator, it is shown that a duality between the existence and nonexistence in an interior/exterior/entire the problems still holds.
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