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Qualitative and Quantitative Analysis of Nonlinear Oscillation of Differential Equations

Research Project

Project/Area Number 09640237
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionFukuoka University

Principal Investigator

KUSANO Takashi  Fukuoka Univ., Fac.of Science, Professor, 理学部, 教授 (70033868)

Co-Investigator(Kenkyū-buntansha) KUROKIBA Masaki  Fukuoka Univ., Fac.of Science, Assistant, 理学部, 助手 (60291837)
TANAKA Naoto  Fukuoka Univ., Fac.of Science, Assoc.Prof., 理学部, 助教授 (00247222)
YOSHIDA Norio  Toyama Univ.Fac.of Science, Professor, 理学部, 教授 (80033934)
NAITO Manabu  Ehime Univ., Fac.of Science, Professor, 理学部, 教授 (00106791)
SAIGO Megumi  Fukuoka Univ., Fac.of Science, Professor, 理学部, 教授 (10040403)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 1998: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1997: ¥1,700,000 (Direct Cost: ¥1,700,000)
Keywordsoscillation theory / nonlinear differential equation / nonlinear Sturm-Liouville operator / oscillatory / zero / singular solution / 振動解.非振動解 / 定性的理論 / 定量解析
Research Abstract

The present research project is devoted to the investigation of the oscillatory behavior of various types of differential equations involving the nonlinear Sturm-Liouville differential operators. The main results obtained are as follows.
(1)We have established Picorie-type identities for the nonlinear Sturm-Liouville operators and have applied them to the study of comparison and oscillation of solutions to half-linear ordinary and partial differential equations. We observe that it has long been unknown whether there is a class of nonlinear differential operators for which a Picone-type identity can be established.
(2)We have developed a theory of generalized trigonometric functions, on the basis of which one can successfully construct an exact nonlinear analogue of the well-known Sturmian theory for linear differential equations. The generalized Prufer transformations thus defined has made it possible to count the number of zeros of nonoscillatory solutions to a certain class of half-linear ordinary differential equations.
(3)We have made a detailed analysis of the asymptotic behavior of positive solutions to some nonlinear Sturm-Liouville equations with singularities. A similar analysis has also been made of differential equations involving singular Sturm-Liouville operators. As an unexpected byproduct we have discovered a new type of singular solution which has never appeared in the literature.
(4)Regarding the oscillation of functional differential equations, we have established (i) a new comparison theorem holding for a special class of non-neutral equations and (ii) an effective criterion for oscillation of neutral equations involving the nonlinear Sturm-Liouville differential operators.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (31 results)

All Other

All Publications (31 results)

  • [Publications] T.Kusano and T.Tanigawa: "Positive solutions to a class of second order differential equations with singular nonlinearities" Applicable Analysis. 69・3-4. 315-331 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Elbert, T.Kusano and M.Naito: "Singular eigenvalue problems for second order linear ordinary differential equations" Archivum Mathematicum(Brno). 34・1. 59-72 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] I.T.Kiguradze and T.Kusano: "On periodical solutions of higher order nonautonomous ordinary differential equations" Differentsial'nye Uravneniya. 印刷中.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] J.Jaros and T.Kusano: "On a class of doubly singular differential equations of second order" Fukuoka University Science Reports. 印刷中.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] J.Jaros, T.Kusano and N.Yoshida: "A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order" Nonlinear Analysis : Theory, Methods and Applications. 印刷中.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano and P.Marusiak: "Asymptotic properties of solutions of second order quasilinear functional differential equuations of neutral type" Mathematica Bohemica. 印刷中.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano, A.Elbert and M.Naito: "Singular eigenvalue problems for second order linear ordinary differential equations" Archivum Mathematicum. 34(1). 59-72 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano and M.Naito: "A singular eigenvalue problem for Sturm-Liouville equations" Differentsial'nye Uravneniya. 34. 303-312 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano and T.Tanigawa: "Positive solutions to a class of second order differential equations with singular nonlinearities" Applicable Analysis. 69(3-4). 315-331 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano, A.Elbert and M.Naito: "Counting the number of zeros of nonoscillatory solutions of second-order half-linear differential equations" Proceedings of the International Scientific Conference of Mathematics(Zilina, June 30-July 3,1998)Zilina. 133-140 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano and N.Yoshida: "Oscillation criteria for a class of functional parabolic equations" Journal of Applied Analysis. 5(1). 1-17 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano and I.T.Kiguradze: "On periodical solutions of higher order nonautonomous ordinary differential equations" Differentsial'nye Uravneniya. (to be published).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano and J.Jaros: "A Picone type identity for second order half-linear differential equations" Acta Mathematica Universitatis Comenianae. (to be publishied).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano, J.Jaros and N.Yoshida: "A Picone type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order" Nonlinear Analysis : Theory, Methods and Applications. (to be published).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano, J.Jaros and P.Marusiak: "Oscillation and nonoscillation theorems for second order quasilinear functional differential equations of neutral type" Advance in Mathematical Sciences and Applications. (to be published).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano and P.Marusiak: "Asymptotic proterties of solutions of second order quasilinear functional differential equations of neutral type" Mathematica Bohemica. (to be published).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano and J.Jaros: "On a class of doubly singular differential equations of second order" Fukuoka University Science Reports. (to be published).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano, J.Kiyomura and M.Naito: "Positive solutions of second order quasilinear ordinary differential equations with general nonlinearities" Studia Scientiarum Mathematicarum Hungarica. (to be published).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano and B.Singh: "Existence of nonoscillatory solutions for nonlinear integro-defferential equations of second order" Fukuoka University Science Reports. (to be published).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano and T.Tanigawa: "Positive solutions to a class of second order differential equations with singular nonlinearities" Applicable Analysis. 69・3-4. 315-331 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] A.Elbert,T.Kusano and M.Naito: "Singular eigenvalue problems for second order linear ordinary differential equations" Archivum Mathematicum(Brno). 34・1. 59-72 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] I.T.Kiguradze and T.Kusano: "On periodical solutions of higher order nonautonomous ordinary differential equations" Differentsial'nye Uravneniya. (印刷中).

    • Related Report
      1998 Annual Research Report
  • [Publications] J.Jaros and T.Kusano: "On class of doubly singular differential equations of second order" Fukuoka University Science Reports. (印刷中).

    • Related Report
      1998 Annual Research Report
  • [Publications] J.Jaros,T.Kusano and N.Yoshida: "A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order" Nonlinear Analysis: Theory,Methods and Applications. (印刷中).

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Kusano and P.Marusiak: "Asymptotic properties of solutions of second order quasilinear functional differential equations of neutral type" Mathematica Bohemica. (印刷中).

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Kusano(共著): "Positive solutions of second order quasilinear functional differential equations with general nonlinearities" Studia Scientiarum Mathematicarum Hungarica. (to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Kusano(共著): "Oscillation and nonoscillations for second order quasilinear functional differential equations of neutral type" Advances in Mathematical Sciences and Applications. (to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Kusano(共著): "Positive entire solutions to nonlinear biharmonic equations in the plane" Journal of computational and Applied Mathematics. (to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Kusano(共著): "A singular eigenvalue problem for second order linear ordinary differential equations" Proceedings of International Symposium on Differential Equations and Mathematical Physics. (to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Kusano(共著): "Singular eigenvalue problems for second order linear differential equations" Proceedings of Conference on Differential Equations and Their Applications(Equadiff 9). (to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Kusano(共著): "Positive solutions to a class of second order differential equations with singular nonlinearities" Applicable Analysis. (to appear).

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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