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

Research Project

Project/Area Number 26400182
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionOkayama 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
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords2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 / 2点境界値問題 / 2点境界理問題 / 国際研究者交流
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.

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。

Report

(6 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (35 results)

All 2019 2018 2017 2016 2015 2014 Other

All Int'l Joint Research (9 results) Journal Article (14 results) (of which Int'l Joint Research: 6 results,  Peer Reviewed: 14 results,  Open Access: 5 results,  Acknowledgement Compliant: 4 results) Presentation (9 results) Remarks (1 results) Funded Workshop (2 results)

  • [Int'l Joint Research] University of Ulsan(韓国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Pontificia Universidad Catolica de Chile/Center for Mathematical Modeling(チリ)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] University of Ulsan(韓国)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] ウルサン大学(韓国)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] ザグレブ大学(クロアチア)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] 清華大学/勤益科技大学(台湾)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] University of Zagreb(クロアチア)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] University of Ulsan(韓国)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] 東北師範大学(中国)

    • Related Report
      2015 Research-status Report
  • [Journal Article] Symmetry-breaking bifurcation for the one-dimensional H?non equation2019

    • Author(s)
      Sim Inbo、Tanaka Satoshi
    • Journal Title

      Communications in Contemporary Mathematics

      Volume: 21 Issue: 01 Pages: 1750097-1750097

    • DOI

      10.1142/s0219199717500973

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Symmetry-breaking bifurcation for the Moore-Nehari differential equation.2018

    • Author(s)
      R. Kajikiya, I. Sim and S. Tanaka
    • Journal Title

      Nonlinear Differential Equations and Applications

      Volume: 25 Issue: 6 Pages: 1-22

    • DOI

      10.1007/s00030-018-0545-3

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Box-counting dimension of oscillatory solutions to the Emden-Fowler equation2018

    • Author(s)
      Kanemitsu Takanao、Tanaka Satoshi
    • Journal Title

      Differential Equations & Applications

      Volume: 10 Issue: 2 Pages: 239-250

    • DOI

      10.7153/dea-2018-10-17

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Rectifiable and nonrectifiable solution curves of half-linear differential systems2018

    • Author(s)
      Yuki Naito, Mervan Pasic, Satoshi Tanaka
    • Journal Title

      Math. Slovaca

      Volume: 68 Issue: 3 Pages: 575-590

    • DOI

      10.1515/ms-2017-0126

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A complete classification of bifurcation diagrams for a class of (p,q)-Laplace equations2018

    • Author(s)
      Kajikiya Ryuji, Sim Inbo, Tanaka Satoshi
    • Journal Title

      Journal of Mathematical Analysis and Applications

      Volume: 462 Issue: 2 Pages: 1178-1194

    • DOI

      10.1016/j.jmaa.2018.02.049

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Box-counting dimension of solution curves for a class of two-dimensional nonautonomous linear differential systems2018

    • Author(s)
      Masakazu Onitsuka, Satoshi Tanaka
    • Journal Title

      Mathematical Communications

      Volume: 23 Pages: 43-60

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Nonrectifiable oscillatory solutions of second order linear differential equations2017

    • Author(s)
      Kanemitsu Takanao, Tanaka Satoshi
    • Journal Title

      Archivum Mathematicum

      Volume: 53 Issue: 4 Pages: 193-201

    • DOI

      10.5817/am2017-4-193

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Symmetry-breaking bifurcation for the one-dimensional Liouville type equation2017

    • Author(s)
      Tanaka Satoshi
    • Journal Title

      Journal of Differential Equations

      Volume: 263 Issue: 10 Pages: 6953-6973

    • DOI

      10.1016/j.jde.2017.07.033

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Characteristic equation for autonomous planar half-linear differential systems2017

    • Author(s)
      Masakazu Onitsuka, Satoshi Tanaka
    • Journal Title

      Acta Mathematica Hungarica

      Volume: 152 Issue: 2 Pages: 336-364

    • DOI

      10.1007/s10474-017-0722-6

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Rectifiability of Solutions for a Class of Two-Dimensional Linear Differential Systems2017

    • Author(s)
      Masakazu Onitsuka, Satoshi Tanaka
    • Journal Title

      Mediterranean Journal of Mathematics

      Volume: 14: 51 Issue: 2 Pages: 1-11

    • DOI

      10.1007/s00009-017-0854-5

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Bifurcation of positive solutions for the one-dimensional (p,q)-Laplace equation2017

    • Author(s)
      Ryuji Kajikiya, Mieko Tanaka, Satoshi Tanaka
    • Journal Title

      Electronic Journal of Differential Equations

      Volume: 2017 Pages: 1-37

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Non-monotone positive solutions of second-order linear differential equations: existence, nonexistence and criteria2016

    • Author(s)
      Mervan Pasic, Satoshi Tanaka
    • Journal Title

      Electronic Journal of Qualitative Theory of Differential Equations

      Volume: 2016 Issue: 93 Pages: 1-25

    • DOI

      10.14232/ejqtde.2016.1.93

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Uniqueness of sign-changing radial solutions for Δu-u+|u|^{p-1}u=0 in some ball and annulus2016

    • Author(s)
      Satoshi Tanaka
    • Journal Title

      Journal of Mathematical Analysis and Applications

      Volume: 439 Issue: 1 Pages: 154-170

    • DOI

      10.1016/j.jmaa.2016.02.036

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Three positive solutions for one-dimensional p-Laplacian problem with sign-changing weight2015

    • Author(s)
      Inbo Sim, Satoshi Tanaka
    • Journal Title

      Applied Mathematics Letters

      Volume: 49 Pages: 42-50

    • DOI

      10.1016/j.aml.2015.04.007

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] Symmetry-breaking bifurcation of positive solutions to the Moore-Nehari differential equation2018

    • Author(s)
      田中 敏、梶木屋 龍治、Inbo Sim
    • Organizer
      日本数学会2018 年度秋季総合分科会
    • Related Report
      2018 Annual Research Report
  • [Presentation] A complete classfication of bifurcation diagrams for a class of (p,q)-Laplace equations2018

    • Author(s)
      梶木屋龍治, Inbo Sim, 田中敏
    • Organizer
      2018年度日本数学会年会
    • Related Report
      2017 Research-status Report
  • [Presentation] Box dimension of solution curves for a class of two-dimensional linear differential systems2018

    • Author(s)
      鬼塚政一, 田中敏
    • Organizer
      2018年度日本数学会年会
    • Related Report
      2017 Research-status Report
  • [Presentation] Characteristic equation for autonomous planar half-linear differential systems2018

    • Author(s)
      田中敏, 鬼塚政一
    • Organizer
      2018年度日本数学会年会
    • Related Report
      2017 Research-status Report
  • [Presentation] Symmetry-breaking bifurcation for the one-dimensional Henon equation2017

    • Author(s)
      田中敏
    • Organizer
      日本数学会2017年度年会
    • Place of Presentation
      首都大学東京
    • Related Report
      2016 Research-status Report
  • [Presentation] Morse index and symmetry-breaking bifurcation for the one-dimensional Liouville type equation2016

    • Author(s)
      田中敏
    • Organizer
      日本数学会2016年度年会
    • Place of Presentation
      筑波大学
    • Year and Date
      2016-03-17
    • Related Report
      2015 Research-status Report
  • [Presentation] Attractivity, rectifiability and non-rectifiability of solutions for two-dimensional linear differential systems2016

    • Author(s)
      鬼塚政一, 田中敏
    • Organizer
      日本数学会2016年度年会
    • Place of Presentation
      筑波大学
    • Year and Date
      2016-03-16
    • Related Report
      2015 Research-status Report
  • [Presentation] Uniqueness of sign-changing radial solutions for the scalar field equation in some ball and annulus2015

    • Author(s)
      田中敏
    • Organizer
      日本数学会2015年度秋季総合分科会
    • Place of Presentation
      京都産業大学
    • Year and Date
      2015-09-14
    • Related Report
      2015 Research-status Report
  • [Presentation] Bifurcation of positive solutions for the one dimensional (p, q)-Laplace equation2014

    • Author(s)
      田中視英子、梶木屋龍治、田中敏
    • Organizer
      日本数学会 2014年度秋季総合分科会
    • Place of Presentation
      広島大学 東広島キンパス
    • Year and Date
      2014-09-26
    • Related Report
      2014 Research-status Report
  • [Remarks] 田中 敏 web page

    • URL

      http://www.xmath.ous.ac.jp/~tanaka/

    • Related Report
      2015 Research-status Report
  • [Funded Workshop] Qualitative Theory on ODEs and their applications to Mathematical Modeling2018

    • Related Report
      2018 Annual Research Report
  • [Funded Workshop] RIMS研究集会 常微分方程式の定性的理論とその周辺 (Qualitative Theory of Ordinary Differential Equations and Related Areas)2016

    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2016-11-16
    • Related Report
      2016 Research-status Report

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Published: 2014-04-04   Modified: 2022-02-22  

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