Israel Journal of Mathematics

, Volume 225, Issue 1, pp 1–33 | Cite as

Well-posedness for mutational equations under a general type of dissipativity conditions

  • Yoshikazu Kobayashi
  • Naoki Tanaka


This paper is concerned with mutational analysis found by Aubin and developed by Lorenz. To extend their results so that they can be applied to quasi-linear evolution equations initiated by Kato, we focus on a mutational framework where for each r > 0 there exists M ≥ 1 such that d(ϑ(t, x), ϑ(t, y)) ≤ Md(x, y) for t ∈ [0, 1] and x, yD r (φ), where ϑ is a transition and Dr(φ) is the revel set of a proper lower semicontinuous functional φ. The setting that the constant M may be larger than 1 plays an important role in applying to quasi-linear evolution equations. In that case, it is difficult to estimate the distance between two approximate solutions to mutational equations. Our strategy is to construct a family of metrics depending on both time and state, with respect to which transitions are contractive in some sense.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. P. Aubin, Viability Theory, ystems & Control: Foundations & Applications, SBirkhäuser Boston, Boston, MA, 1991.zbMATHGoogle Scholar
  2. [2]
    J. P. Aubin, Mutational equations in metric spaces, Set-Valued Analysis 1 (1993), 3–46.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    J. P. Aubin, Mutational and Morphological Analysis, Tools for Shape Evolution and Morphogenesis, Systems & Control: Foundations & Applications, Birkhaüser Boston, Boston, MA, 1999.CrossRefzbMATHGoogle Scholar
  4. [4]
    P. Gwiazda, T. Lorenz and A. M. Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, Journal of Differential Equations 248 (2010), 2703–2735.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    T. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Archive for Rational Mechanics and Analysis 63 (1976), 273–294 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    T. Kato, Linear evolution equations of “hyperbolic” type. II, Journal of the Mathematical Society of Japan 25 (1973), 648–666.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974; Dedicated to Konrad Jörgens), Lecture Notes in Mathematics, Vol. 448, Springer, Berlin, 1975, pp. 25–70.Google Scholar
  8. [8]
    Y. Kobayashi and N. Tanaka, Semigroups of Lipschitz operators, Advances in Differential Equations 6 (2001), 613–640.MathSciNetzbMATHGoogle Scholar
  9. [9]
    V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, Vol. I: Ordinary Differential Equations, Mathematics in Science and Engineering, Vol. 55-I, Academic Press, New York–London, 1969.zbMATHGoogle Scholar
  10. [10]
    V. Lakshmikantham, R. Mitchell and R. Mitchell, Differential equations on closed subsets of a Banach space, Transactions of the American Mathematical Society 220 (1976), 103–113.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    T. Lorenz, Mutational Analysis, A Joint Framework for Cauchy Problems in and beyond Vector Spaces, Lecture Notes in Mathematics, Vol. 1996, Springer-Verlag, Berlin, 2010.Google Scholar
  12. [12]
    R. H. Martin, Jr., Differential equations on closed subsets of a Banach space, Transactions of the American Mathematical Society 179 (1973), 399–414.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    M. Nagumo, Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen, Proceedings of the Physico-Mathematical Society of Japan 24 (1942), 551–559.zbMATHGoogle Scholar
  14. [14]
    H. Okamura, Condition nécessaire et suffisante remplie par les équations différentielles ordinaires sans points de Peano, Memoirs of the College of Science, Kyoto Imperial University. Series A. 24 (1942), 21–28.zbMATHGoogle Scholar
  15. [15]
    T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Publications of the Mathematical Society of Japan, Vo. 9, The Mathematical Society of Japan, Tokyo, 1966.zbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and EngineeringChuo UniversityTokyoJapan
  2. 2.Department of Mathematics, Faculty of ScienceShizuoka UniversityShizuokaJapan

Personalised recommendations