Precise asymptotic analysis of solutions of nonlinear differential equations by means of regular variation: The theoretical face and back sides for oscillation theory

• Project/Area Number: 23540218
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kumamoto University
• Principal Investigator: Tanigawa Tomoyuki 熊本大学, 教育学部, 准教授 (10332008)
• Project Period (FY): 2011-04-28 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 微分方程式論 / 振動理論 / 非振動解の漸近挙動 / 正則変動関数論 / 微分方程式 / 国際研究者交流 / 多国籍

Outline of Final Research Achievements

The purpose of this research subject is devoted to the asymptotic analysis of the nonoscillatory behavior of several types of the nonlinear differential equations. Our main results obtained are as belows.
(1) The existence and the asymptotic behavior for the large value of the variable of the positive solutions of generalized Thomas-Fermi equation are proved. (2) We establish a sharp condition of the existence of generalized regularly varying functions (in the sense of Karamata) of self-adjoint functional differential equation. (3) We devote to the asymptotic analysis of a class of the third order sublinear differential equation. (4) We demonstrate that the method of regular variation can be effectively applied to fourth order quasilinear differential equations. (5) We show that an application of the theory of regular variation gives the possibility of determining the existence and precise asymptotic behavior of positive solutions of the third order nonlinear differential equation

Research Products

• [Journal Article] Asymptotic analysis of positive decreasing solutions of a class of systems of second order nonlinear differential equations in the framework of regular variation (2015)
  • Author(s): Ｊ． Ｊａｔｏｓ， Ｔ． Ｋｕｓａｎｏ ａｎｄ Ｔ． Ｔａｎｉｇａｗａ
  • Journal Title: The Australian Journal of Mathematical Analysis and Applications Volume: 12
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Existence and asymptotic behavior of strongly monotone solutions of nonlinear differential equations (2015)
  • Author(s): Ｔｏｍｏｙｕｋｉ Ｔａｎｉｇａｗａ
  • Journal Title: Differential Equations and Applications Volume: 7 Issue: 1 Pages: 79-91
  • DOI: 10.7153/dea-07-06
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Regularly varying solutions of half-linear differential equations with retarded and advanced arguments (2015)
  • Author(s): J. Manojlovic and T. Tanigawa
  • Journal Title: Mathematica Slovaca Volume: 65
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Asymptotic analysis of positive decreasing solutions of a class of systems of second order nonlinear differential equations in the framework of regular variation. (2015)
  • Author(s): J. Jaros, T. Kusano and T. Tanigawa
  • Journal Title: The Australian Journal of Mathematical Analysis and Applications Volume: 12
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Existence and asymptotic behavior of strongly monotone solutions of nonlinear differential equations. (2015)
  • Author(s): Tomoyuki Tanigawa
  • Journal Title: Differential Equations and Applications Volume: 7 Pages: 79-91
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Asymptotic analysis of positive solutions of third order nonlinear differential equations (2014)
  • Author(s): J. Jaros, T. Kusano and T. Tanigawa
  • Journal Title: Hiroshima Mathematical Journal Volume: 44
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Existence and asymptotic behavior of positive solutions of fourth order quasilinear differential equations (2013)
  • Author(s): T. Kusano, J. Manojlovic and T. Tanigawa
  • Journal Title: Taiwanese Journal of Mathematics Volume: 17 Pages: 999-1030
  • Peer Reviewed
• [Journal Article] Existence and precise asymptotics of positive solutions for a class of nonlinear differential equations of the third order (2013)
  • Author(s): J. Jaros, T. Kusano and T. Tanigawa
  • Journal Title: Georgian Mathematical Journal Volume: 20 Pages: 493-531
  • Peer Reviewed
• [Journal Article] Asymptotic analysis of positive solutions of third order nonlinear differential equations (2013)
  • Author(s): J. Jaros, T. Kusano and T. Tanigawa
  • Journal Title: Hiroshima Mathematical Journal Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Generalized regularly varying solutions of second order nonlinear differential equations with deviating arguments (2012)
  • Author(s): Tomoyuki Tanigawa
  • Journal Title: Memoirs on Differential Equations and Mathematical Physics Volume: 57
  • Peer Reviewed
• [Journal Article] Asymptotic analysis of positive solutions of a class of third order nonlinear differential equations in the framework of regular variation (2012)
  • Author(s): J. Jaros, T. Kusano and T. Tanigawa
  • Journal Title: Mathematische Nachrichten Volume: 286 Issue: 2-3 Pages: 205-223
  • DOI: 10.1002/mana.201100296
  • Peer Reviewed
• [Journal Article] An asymptotic analysis of positive solutions of generalized Thomas--Fermi differential equations -- The sub-half-linear case (2012)
  • Author(s): Takasi Kusano, Vojislav Maric and Tomoyuki Tanigawa
  • Journal Title: Nonlinear Analysis: Theory, Methods & Applications Volume: 75
  • Peer Reviewed
• [Journal Article] Sharp oscillation criteria for a class of fourth order nonlinear differential equations (2011)
  • Author(s): Takasi Kusano, Jelena Manojlovic and Tomoyuki Tanigawa
  • Journal Title: The Rocky Mountain Journal of Mathematics Volume: 41
  • Peer Reviewed
• [Presentation] 混合型2階半分線形関数微分方程式の正値解の漸近挙動について (2016)
  • Author(s): 谷川智幸
  • Organizer: 日本数学会年会
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-16
• [Presentation] Asymptotic behavior of positive solutions of second order half-linear differential equations with deviating arguments of mixed type (2015)
  • Author(s): Tomoyuki Tanigawa
  • Organizer: International Workshop on the Qualitative Theory of Differential Equations, QUALITDE 2015
  • Place of Presentation: Tbilisi, Georgia
  • Year and Date: 2015-12-27
  • Int'l Joint Research / Invited
• [Presentation] Existence and asymptotic behavior of strongly monotone solutions of nonlinear differential equations (2015)
  • Author(s): Tomoyuki Tanigawa
  • Organizer: The Equadiff 2015
  • Place of Presentation: Lyon, France
  • Year and Date: 2015-07-06
  • Int'l Joint Research
• [Presentation] Asymptotic analysis of strongly monotone solutions of nonlinear differential equations (2015)
  • Author(s): Tomoyuki Tanigawa
  • Organizer: Conference on Differential \& Difference Equations and Applications
  • Place of Presentation: Lisbon, Portugal
  • Year and Date: 2015-05-18
  • Int'l Joint Research / Invited
• [Presentation] 4階劣半分線形微分方程式の正値解の存在について (2015)
  • Author(s): 谷川智幸
  • Organizer: 日本数学会年会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-21 – 2015-03-24
• [Presentation] ある4 階非線形微分方程式の正則変動関数解の存在について (2015)
  • Author(s): 谷川智幸
  • Organizer: 常微分方程式ワークショップ
  • Place of Presentation: 愛媛大学
  • Year and Date: 2015-03-10
• [Presentation] 正則変動関数論の微分方程式への応用 (2015)
  • Author(s): 谷川智幸
  • Organizer: 第7回半田山微分方程式セミナー
  • Place of Presentation: 岡山理科大学
  • Year and Date: 2015-02-24
  • Invited
• [Presentation] 3 階 Emden - Fowler 型微分方程式の正値解の漸近挙動について (2014)
  • Author(s): 谷川智幸
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-25 – 2014-09-28
• [Presentation] 遅れの変数を含む２階半分線形微分方程式の零でない指数の正則変動関数解の存在について (2014)
  • Author(s): 谷川智幸
  • Organizer: 振動理論ワークショップ -- 金沢 2013
  • Place of Presentation: 金沢大学
• [Presentation] Asymptotic analysis of positive decreasing solutions of a class of systems of second order nonlinear differential equations in the framework of regular variation (2013)
  • Author(s): Tomoyuki Tanigawa
  • Organizer: The Equadiff 13 conference 2013
  • Place of Presentation: Prague, Czech Republic
  • Invited
• [Presentation] Asymptotic behavior of decreasing solutions of a class of systems of second order nonlinear differential equations (2013)
  • Author(s): Tomoyuki Tanigawa
  • Organizer: Colloquium on Differential Equations
  • Place of Presentation: Nis, Serbia
  • Invited
• [Presentation] ２階非線形微分方程式系の正値減少解について (2013)
  • Author(s): 谷川智幸
  • Organizer: 岡山理科大学における微分方程式セミナー
  • Place of Presentation: 岡山理科大学
• [Presentation] 進みと遅れの変数を含む２階半分線形関数微分方程式の一般化された正則変動関数解の存在について (2013)
  • Author(s): 谷川智幸
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 愛媛大学
• [Presentation] ある２階非線形微分方程式系の正値減少解の漸近挙動について (2013)
  • Author(s): 谷川智幸
  • Organizer: 第174回 AP Seminar (研究集会「富山解析セミナー2013」)
  • Place of Presentation: 富山大学
• [Presentation] ある３階非線形微分方程式の正値解の漸近挙動について (2013)
  • Author(s): 谷川智幸
  • Organizer: 振動理論ワークショップ ー 松山 2013
  • Place of Presentation: 愛媛大学
  • Invited
• [Presentation] Generalized regularly varying solutions of half-linear functional differential equations with retarded and advanced arguments (2012)
  • Author(s): Tomoyuki Tanigawa
  • Organizer: Conference on Differential and Difference Equations and Applications
  • Place of Presentation: スロバキア（ジリナ）
  • Invited
• [Presentation] 進みと遅れの変数をもつ2 階半分線形関数微分方程式の正値解について (2012)
  • Author(s): 谷川智幸
  • Organizer: 愛知教育大学における微分方程式セミナー
  • Place of Presentation: 愛知教育大学
  • Invited
• [Presentation] 微分方程式の解の漸近挙動とKaramata 関数について (2012)
  • Author(s): 谷川智幸
  • Organizer: 第21 回岐阜数理科学セミナー
  • Place of Presentation: 岐阜大学
  • Invited
• [Presentation] Precise asymptotic behavior of positive solutions of generalized Thomas -- Fermi differential equations (2011)
  • Author(s): Tomoyuki Tanigawa
  • Organizer: Progress in Qualitative Theory of Functional Equations(招待講演)
  • Place of Presentation: Research Institute for Mathematical Sciences Conference, Kyoto University
• [Presentation] 一般化された Thomas - Fermi 型微分方程式の正則変動解の漸近解析について
  • Author(s): 谷川智幸
  • Organizer: 岐阜大学における微分方程式セミナー(招待講演)
  • Place of Presentation: 岐阜大学
• [Presentation] 一般化された Thomas - Fermi 型微分方程式の解の漸近解析について
  • Author(s): 谷川智幸
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 信州大学
• [Presentation] ある非半分線形微分方程式の解の漸近挙動について
  • Author(s): 谷川智幸
  • Organizer: 富山解析セミナー 2011(招待講演)
  • Place of Presentation: 富山大学