Oscillation Theory for Nonlinear Differential Equations and Its Applications to Elliptic Equations

• Project/Area Number: 19740074
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Osaka Prefecture University
• Principal Investigator: YAMAOKA Naoto Osaka Prefecture University, 工学研究科, 准教授 (90433789)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥2,920,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥420,000); Fiscal Year 2009: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2007: ¥1,100,000 (Direct Cost: ¥1,100,000)
• Keywords: 常微分方程式 / 楕円型方程式 / 振動理論 / 非線形摂動項 / 比較定理 / 相平面解析 / 時間遅れ / 関数方程式論 / 定性的理論 / 振動定数 / 振動解 / 正値解

Research Abstract

The purpose of this research project is to study oscillation problems for ordinary differential equations and elliptic equations with p-Laplacian. The main results are as follows: (1) Oscillation and non-oscillation criteria are given for all nontrivial solutions of second-order damped nonlinear differential equations with one-dimensional p-Laplacian. (2) A sufficient condition is obtained for half-linear differential equations with delay nonlinear perturbations to have a non-oscillatory solution. (3) We establish a new comparison theorem on the oscillation of solutions of nonlinear differential equations with one-dimensional p-Laplacian. (4) We consider radial solutions of elliptic equations with p-Laplacian, and give sufficient conditions for the equations to have a positive solution.

Research Products

• [Journal Article] A comparison theorem and oscillation criteria for second-order nonlinear differential equations (2010)
  • Author(s): N. Yamaoka
  • Journal Title: Appl. Math. Lett. 23 Pages: 902-906
  • Peer Reviewed
• [Journal Article] A nonoscillation theorem for half-linear differential equations with delay nonlinear perturbations (2009)
  • Author(s): N. Yamaoka
  • Journal Title: Diff. Eq. Appl. 1 Pages: 209-217
  • Peer Reviewed
• [Journal Article] p-Laplacianをもつ非線形微分方程式の振動に対する比較定理, 関数方程式のダイナミクスと数理モデル (2009)
  • Author(s): N. Yamaoka
  • Journal Title: 数理解析研究所講究録 1637 Pages: 25-31
• [Journal Article] Comparison theorems for oscillation of nonlinear differential equations with p-Laplacian (2009)
  • Author(s): N.Yamaoka
  • Journal Title: 数理解析研究所講究録 1637 Pages: 25-31
• [Journal Article] A nonoscillation theorem for half-linear differential equations with delay nonlinear perturbations (2009)
  • Author(s): N. Yamaoka
  • Journal Title: Differential Equations & Applications 1 Pages: 209-217
  • Peer Reviewed
• [Journal Article] Oscillation constants of nonlinear differential equations with delay, 現象からの関数方程式 (2007)
  • Author(s): N. Yamaoka
  • Journal Title: 数理解析研究所講究録 1547 Pages: 87-93
• [Journal Article] Oscillation criteria for second-order damped nonlinear differential equations with p-Laplacian (2007)
  • Author(s): N. Yamaoka
  • Journal Title: J. Math. Anal. Appl. 325 Pages: 932-948
  • Peer Reviewed
• [Journal Article] Oscillation criteria for second-order damped nonlinear differential equations with p-Laplacian (2007)
  • Author(s): N.Yamaoka
  • Journal Title: Journal of Mathematical Analysis and Applications 325 Pages: 932-948
  • Peer Reviewed
• [Journal Article] Oscillation constants of nonlinear differential equations with delay (2007)
  • Author(s): N.Yamaoka
  • Journal Title: 京都大学数理解析研究所講究録 1547 Pages: 87-93
• [Journal Article] A comparison theorem and oscillation criteria for second-order nonlinear differential equations
  • Author(s): N.Yamaoka
  • Journal Title: Applied Mathematics Letters (印刷中)
  • Peer Reviewed
• [Presentation] Comparison theorems for oscillation of second-order nonlinear differential equations (2009)
  • Author(s): N.Yamaoka
  • Organizer: Equadiff 12
  • Place of Presentation: Masaryk University (Brno, Czech Republic)
  • Year and Date: 2009-07-20
• [Presentation] 1次元p-Laplacianをもつ非線形微分方程式の振動に対する比較定理 (2009)
  • Author(s): 山岡直人
  • Organizer: 日本数学会2009年度年会
  • Place of Presentation: 東京大学
  • Year and Date: 2009-03-26
• [Presentation] 1次元p-Laplacianをもつ非線形微分方程式の振動に対する比較定理 (2009)
  • Author(s): 山岡直人
  • Organizer: 日本数学会2009年度年会函数方程式論分科会
  • Place of Presentation: 東京大学
  • Year and Date: 2009-03-26
• [Presentation] 2階準線形楕円型方程式の振動条件について (2009)
  • Author(s): 山岡直人
  • Organizer: 研究集会「微分方程式の定性的理論ワークショップ」
  • Place of Presentation: 大阪府立大学
  • Year and Date: 2009-03-03
• [Presentation] 2階準線形楕円型方程式の振動条件について (2009)
  • Author(s): 山岡直人
  • Organizer: 研究集会「振動理論ワークショップ-松山2009」
  • Place of Presentation: 愛媛大学
  • Year and Date: 2009-02-07
• [Presentation] Comparison theorems for oscillation of second-order nonlinear differential equations, Equadiff 12 (2009)
  • Author(s): N. Yamaoka
  • Organizer: Brno, Czech Republic
  • Place of Presentation: Masaryk University
• [Presentation] p-Laplacianをもつ非線形微分方程式の振動に対する比較定理 (2008)
  • Author(s): 山岡直人
  • Organizer: 研究集会「関数方程式のダイナミクスと数理モデル」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-11-04
• [Presentation] p-Laplacianをもつ非線形微分方程式の振動に対する比較定理 (2008)
  • Author(s): 山岡直人
  • Organizer: 研究集会「札幌医科大学における微分方程式セミナー(通算第31回)」
  • Place of Presentation: 札幌医科大学
  • Year and Date: 2008-09-01
• [Presentation] 非線形摂動を伴う半分線形微分方程式の解の振動について (2008)
  • Author(s): 山岡直人
  • Organizer: 日本数学会2008年度年会
  • Place of Presentation: 近畿大学
  • Year and Date: 2008-03-23
• [Presentation] 非線形摂動を伴う半分線形微分方程式の解の振動について (2008)
  • Author(s): 山岡 直人
  • Organizer: 日本数学会2 0 0 8年度年会 函数方程式論分科会
  • Place of Presentation: 近畿大学
  • Year and Date: 2008-03-23
• [Presentation] 周期係数をもつ半分線形微分方程式の解の振動について (2008)
  • Author(s): 山岡直人
  • Organizer: 研究集会「微分方程式の定性的理論ワークショップ」
  • Place of Presentation: 徳島大学
  • Year and Date: 2008-03-01
• [Presentation] Oscillation problems for nonlinear differential equations with p-Laplacian and its applications to elliptic equations (2008)
  • Author(s): N. Yamaoka
  • Organizer: Workshop on Stationary Problems in Nonlinear Partial Differential Equations
  • Place of Presentation: Kobe University, Kobe, Japan
• [Presentation] Oscillation problems for half-linear differential equations with nonlinear perturbations (2008)
  • Author(s): N. Yamaoka
  • Organizer: World Congress of Nonlinear Analysts (WCNA 2008)
  • Place of Presentation: Hyatt Grand Cypress Resort in Orlando, Florida, USA
• [Presentation] 自己随伴非線形微分方程式の臨界減衰について (2007)
  • Author(s): 山岡直人
  • Organizer: 研究集会「微分方程式の総合的研究」
  • Place of Presentation: 東京大学
  • Year and Date: 2007-12-14
• [Presentation] 自己随伴非線形微分方程式の減衰解の漸近挙動について (2007)
  • Author(s): 山岡直人
  • Organizer: 第113回広島数理解析セミナー
  • Place of Presentation: 広島大学
  • Year and Date: 2007-12-07
• [Presentation] 非線形摂動項を伴う半分線形微分方程式の振動問題について (2007)
  • Author(s): 山岡直人
  • Organizer: 研究集会「関数方程式論におけるモデリングと複素解析」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2007-11-08
• [Presentation] 非線形摂動項を伴う自励系半分線形微分方程式の振動問題 (2007)
  • Author(s): 山岡直人
  • Organizer: 研究集会「上越教育大学における微分方程式セミナー(通算第30回)」
  • Place of Presentation: 上越教育大学
  • Year and Date: 2007-08-29
• [Presentation] 非線形摂動項を伴う自励系半分線形微分方程式の非振動条件 (2007)
  • Author(s): 山岡直人
  • Organizer: 研究集会「微分方程式湖畔セミナーin松江2007」
  • Place of Presentation: 島根大学
  • Year and Date: 2007-08-03