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2005 Fiscal Year Final Research Report Summary

Discrete functional equations in complex domains

Research Project

Project/Area Number 16540202
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionNippon Institute of Technology

Principal Investigator

ISHIZAKI Katsuya  Nippon Institute of Technology, Department of Technology, Associate Professor, 工学部, 助教授 (60202991)

Co-Investigator(Kenkyū-buntansha) MORI Seiki  University of Yamagata, Department of Science, Professor, 理学部, 教授 (80004456)
SHIMOMURA Shun  University of Keio, Department of Science, Professor, 理工学部, 教授 (00154328)
MOROSAWA Shunsuke  University of Kochi, Department of Science, Professor, 理学部, 教授 (50220108)
TOHGE Kazuya  University of Kanazawa, Department of Technology, Associate Professor, 工学部, 助教授 (30260558)
SAWADA Kazunari  Tokyo Metropolitan Institute of Technology, Department of Technology, Associate Professor, 一般科目, 助教授 (10270232)
Project Period (FY) 2004 – 2005
KeywordsDifference equations / q-difference equations / The Schroeder equation / Nevanlinna theory / Complex dynamics / Uniqueness theory / Algebroid functions / The Painleve equation
Research Abstract

By means of the value distribution theory, we investigated ordinary differential equations and discrete functional equations in complex domains, in particular, linear difference equations having polynomial coefficients and the Schroeder functional equations. We obtain results on Malmquist-Yoshida type theorems for nonlinear differential equations in connection with complex dynamics theory, and results on linear differential equation of the second order with elliptic coefficients. We also constructed Wiman-Valiron theory for binomial series in order to treat linear difference equations. The Schroeder equations are considered from the point of view of the connections between the Nevanlinna theory and the complex dynamics theory. In particular, we gave a proof of the result on Borel directions of the Schroeder functions and the Julia set of the rational functions from which the Schroeder functions are generated. Below we report the researches by each investigator.
Mori obtained uniqueness … More theorems on some complex domains which are bounded or countable.
Shimomura investigated Painleve transcendents. He obtained the lower estimate on meromorphic solutions of higher order Painleve equations. Modified type of Painleve equations (III), and (V) are considered. He also gave nontrivial examples on nonlinear differential equations having the Painleve properties.
Morosawa considered the parameter space of complex dynamics on the complex error functions. He obtained the topological properties of the Julia set of the hyperbolic complex error function.
Fermat type functional equations are treated by Tohge. With Professor Gundersen (New Orleans), he found a new entire function satisfying a Fermat type equation. He also investigated the value distribution theory in angular domains.
Sawada obtained an algebroid function with three sheets on which analytic functions have deficiencies restricted by some given conditions. He also treated meromorphic solutions of some algebraic differential equations. Less

  • Research Products

    (40 results)

All 2006 2005 2004 Other

All Journal Article (40 results)

  • [Journal Article] Borel and Julia directions of meromorphic Schroeder functions II2006

    • Author(s)
      Ishizaki, K., N.Yanagihara
    • Journal Title

      Arch. Math. (発表予定)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Singular directions of meromorphic solutions of some non -autonomous Schroeder equations2006

    • Author(s)
      Ishizaki, K., N.Yanagihara
    • Journal Title

      Advanced Studies of Pure Mathematics, XX, 2005, Potential Theory in Matsue. (発表予定)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A class of differential elations of Pl-type with the quasi-Painleve' property2006

    • Author(s)
      Shimomura, S.
    • Journal Title

      Ann. Mat Pura Appl. (発表予定)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Remarks on deficiencies for meromorphic Schroeder functions2005

    • Author(s)
      Ishizaki, K., N.Yanagihara
    • Journal Title

      Proceedings of The 12th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications.

      Pages: 105-112

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Borel and Julia directions of meromorphic Schroeder functions2005

    • Author(s)
      Ishizaki, K., N.Yanagihara
    • Journal Title

      Math. Proc. Camb. Phi1. Soc. 139(1)

      Pages: 139-147

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Deficiencies of meromorphic mappings for hypersurfaces2005

    • Author(s)
      Aihara, Y., S.Mori
    • Journal Title

      J. Math. Soc. Japan 57

      Pages: 233-258

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Uniqueness of meromorphic functions2005

    • Author(s)
      Kiong, W., W.Lin, S.Mori
    • Journal Title

      Sci. Math. Jpn. 62

      Pages: 305-315

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Meromorphic solutions of Riccati doubly differential equations with doubly periodic coefficients2005

    • Author(s)
      Shimomura, S.
    • Journal Title

      J.Math. Anal Appl. 304

      Pages: 644-651

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On certain nonlinear differential equations2005

    • Author(s)
      Shimomura, S.
    • Journal Title

      Proceedings of the 12th International Conference on Finite or Infinite or Infinite Dimensional Complex Analysis and Applications

      Pages: 299-304

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Dynamics of entire functions with two singular values2005

    • Author(s)
      Taniguchi, M., S.Morosawa
    • Journal Title

      数理解析研究所講究録 1447

      Pages: 130-137

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Some examples of holomorphic curves in Pn(C) extremal to Cartan's defect relation.2005

    • Author(s)
      Tohge, K.
    • Journal Title

      Proc. of 12th Int. Conference on Finite or Infinite Dimensional Complex Analysis and Applications

      Pages: 387-395

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Singular directions of meromorphic solutions of some non-autonomous Schroeder equations2005

    • Author(s)
      Ishizaki, K, N.Yanagihara
    • Journal Title

      Advanced Studies of Pure Mathematics, Potential Theory in Matsue. XX(To appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Remarks on deficiencies for meromorphic Schroeder functions2005

    • Author(s)
      Ishizaki, K, N.Yanagihara
    • Journal Title

      Proceedings of The 12th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications

      Pages: 105-112

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Borel and Julia directions of meromorphic Schroeder functions2005

    • Author(s)
      Ishizaki, K, N.Yanagihara
    • Journal Title

      Math.Proc.Camb.Phil.Soc. 139(1)

      Pages: 139-147

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Deficiencies of meromorphic mapping for hypersurfaces2005

    • Author(s)
      Aihara, Y., S.Mori
    • Journal Title

      J.Math.Soc.Japan 57

      Pages: 233-258

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Uniqueness of meromorphic functions, Sci.Math.Jpn.2005

    • Author(s)
      Xiong, W., W.Lin, S.Mori
    • Journal Title

      Uniqueness of meromorphic functions 62

      Pages: 305-315

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Meromorphic solutions of Riccati differential equations with doubly periodic coefficients2005

    • Author(s)
      Shimomura, S.
    • Journal Title

      J.Math.Anal.Appl. 304

      Pages: 644-651

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On certain nonlinear differential equations2005

    • Author(s)
      Shimomura, S.
    • Journal Title

      Proceedings of the 12th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications

      Pages: 299-304

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Dynamics of entire functions with two singular values2005

    • Author(s)
      Taniguchi, M., S.Morosawa
    • Journal Title

      Surikaisekiken Kokyuroku 1447

      Pages: 130-137

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Some examples of holomorphic curves in Pn(C) extremal to Cartan's defect relation2005

    • Author(s)
      Tohge, K.
    • Journal Title

      Proc.of 12th Int.Conference on Finite or Infinite Dimensional Complex Analysis and Applications

      Pages: 387-395

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Deficiency for meromorphic solutions of the Schroeder equations2004

    • Author(s)
      Ishizaki, K., N.Yanagihara
    • Journal Title

      Complex Variables Theory Appl. 49

      Pages: 539-548

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Woman-Valiron method for difference equations2004

    • Author(s)
      Ishizaki, K., N.Yanagihara
    • Journal Title

      Nagoya Math. J. 175

      Pages: 75-102

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Meromorphic mappings and deficiencies2004

    • Author(s)
      Mori, S.
    • Journal Title

      Advanced Studies in Pure Mathematics, Complex analysis in Several Variables, Mathematical society of Japan

      Pages: 237-242

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Growth of modified Painleve transcendents of the fifth and the third kind2004

    • Author(s)
      Shimomura, S.
    • Journal Title

      Forum Math. 16

      Pages: 231-247

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Lower estimates for the growth of the fourth and the second Paialeve transcendents2004

    • Author(s)
      Shimomura, S.
    • Journal Title

      Proc. Edinburgh Math. 47

      Pages: 231-249

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Poles and alpha-points of meromorphic solutions of the first Pailileve hierarchy2004

    • Author(s)
      Shimomura, S.
    • Journal Title

      Publ. Res. Inst. Math. Sci. 40

      Pages: 471-485

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On the number of poles of the first Painleve transcendents and higher order analogues2004

    • Author(s)
      Shimomura, S.
    • Journal Title

      Proceedings of Symposium on Complex Differential and Functional Equations, Joensuu University. 6

      Pages: 149-154

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A certain expression of the first Painlev'e hierarchy2004

    • Author(s)
      Shimomura, S.
    • Journal Title

      Proc. Japan Acad. Ser. A 80

      Pages: 105-109

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Fatou components whose boundaries have a common curve2004

    • Author(s)
      Morosawa, S.
    • Journal Title

      Fund.Math. 183

      Pages: 47-57

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Entire and meromorphic solutions of f^5+g^5+h^5=12004

    • Author(s)
      Gundersen, G. G., K.Tohge
    • Journal Title

      Proceedings of Symposium on Complex Differential and Functional Equations, Joensuu University. 6

      Pages: 57-67

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Borel and Julia directions of meromorphic Schroeder functions2004

    • Author(s)
      Ishizaki, K, N.Yanagihara
    • Journal Title

      Complex Variables Theory Appl. 49

      Pages: 539-548

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Meromorphic mappings and deficiencies2004

    • Author(s)
      Mori, S.
    • Journal Title

      Advanced Studies in Pure Mathematics, Complex Analysis in Several Variables, Mathematical Society of Japan

      Pages: 237-242

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Lower estimates for the growth of the fourth and the second Painleve transcendents2004

    • Author(s)
      Shimomura, S.
    • Journal Title

      Proc.Edinburgh Math.Soc. 47

      Pages: 231-249

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Poles and alpha-points of meromorphic solutions of the first Painleve hierarchy2004

    • Author(s)
      Shimomura, S.
    • Journal Title

      Publ.Res.Inst.Math.Sci. 40

      Pages: 471-485

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On the number of poles of the first Painleve transcendents and higher order analogues2004

    • Author(s)
      Shimomura, S.
    • Journal Title

      Proceedings of Symposium on Complex Differential and Functional Equations, Joensuu University 6

      Pages: 149-154

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A certain expression of the first Painleve hierarchy2004

    • Author(s)
      Shimomura, S.
    • Journal Title

      Proc.Japan Acad.Ser.A 80

      Pages: 105-109

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Borel and Julia directions of meromorphic Schroeder functions II

    • Author(s)
      Ishizaki, K, N.Yanagihara
    • Journal Title

      Arch.Math. (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Wiman-Valiron method for difference equations

    • Author(s)
      Ishizaki, K, N.Yanagihara
    • Journal Title

      Nagoya Math.J. 175

      Pages: 75-102

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A class of differential equations of PI-type with the quasi-Painleve property

    • Author(s)
      Shimomura, S.
    • Journal Title

      Ann.Mat.Pura Appl. (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Entire and meromorphic solutions of f^5+g^5+h^5=1

    • Author(s)
      Gundersen, G.G., K.Tohge
    • Journal Title

      Proceedings of Symposium on Complex Differential and Functional Equations, Joensuu University 6

      Pages: 57-67

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2007-12-13  

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