2018 Fiscal Year Final Research Report

The number of solutions and the length and the fractal dimension of oscillatory solutions of two point boundary problems

Research Project

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Project/Area Number	26400182
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Multi-year Fund
Section	一般
Research Field	Mathematical analysis
Research Institution	Okayama University of Science
Principal Investigator	Tanaka Satoshi 岡山理科大学, 理学部, 教授 (90331959)
Research Collaborator	Onitsuka Masakazu Naito Yuki Kajikiya Ryuji Tanaka Mieko Kanemitsu Takanao Shioji Naoki Watanabe Kohtaro Pasic Mervan Sim Inbo Wu Fentao Wang Shin-Hwa Hung Kuo-Chih Manasevich Raul Garcia-Huidobro Marta
Project Period (FY)	2014-04-01 – 2019-03-31
Keywords	２点境界問題 / 正値解 / 符号変化する解 / 分岐 / 解曲線の長さ / フラクタル次元 / ボックス次元
Outline of Final Research Achievements	The number of solutions and the length and the fractal dimension of oscillatory solutions of two point boundary problems are studied, and then the following results were established. Sufficient conditions for the existence of three positive solutions to two point boundary problem with a sign-changing weight function and the one-dimensional p-Lalpacian were obtained. Results on the number of solutions to the autonomous two point boundary problem with (p,q)-Lalpacian were obtained. The symmetry-breaking bifurcations for the one-dimensional Liouville type equation and the one-dimensional Henon equation were found. Results on the non-rectifiability and the box-counting dimension of solution curves of two-dimensional non-autonomous linear and half-linear differential systems were obtained.
Free Research Field	微分方程式論
Academic Significance and Societal Importance of the Research Achievements	本研究のテーマである２点境界値問題は、それ自身が微分方程式論のなかで重要な問題であるが、偏微分方程式の研究でもしばしばあらわれるもので、その解の個数を知ることは基礎的かつ重要な問題である。２点境界問題の解の存在・非存在に関してこれまで膨大な量の結果が得られている一方で、その解の厳密な個数を調べることは、問題が単純な形であっても非常な困難を伴うことが多い。また、振動解のグラフの有限長性とフラクタル次元についての研究は、つい最近始まった独創的な研究である。本研究により、以上のような問題の一部が解決された。