Positivity

, Volume 22, Issue 1, pp 105–138 | Cite as

Measures under the flat norm as ordered normed vector space

  • Piotr Gwiazda
  • Anna Marciniak-Czochra
  • Horst R. Thieme
Article
  • 79 Downloads

Abstract

The space of real Borel measures \(\mathcal {M}(S)\) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone \(\mathcal {M}_+(S)\) of nonnegative measures, is considered from an ordered normed vector space perspective in order to apply the well-developed theory of this area. The flat norm is considered in place of the variation norm because subsets of \(\mathcal {M}_+(S)\) are compact and semiflows on \(\mathcal {M}_+(S)\) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because \(\mathcal {M}(S)\) is rarely complete and \(\mathcal {M}_+(S)\) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on \(\mathcal {M}_+(S)\) and continuous semiflows. Both topics prepare for a dynamical systems theory on \(\mathcal {M}_+(S)\).

Keywords

Measures Ordered normed vector space Homogeneous maps Spectral radius Krein–Rutman theorem Dynamical systems 

Mathematics Subject Classification

Primary 28A33 28C15 46E27 47B65 47D06 47H07 47J10 58C40 Secondary 47N60 

Notes

Acknowledgements

The authors thank Azmy Ackleh, Stephan Luckhaus, and an anonymous referee for useful comments.

References

  1. 1.
    Ackleh, A.S., Cleveland, J., Thieme, H.R.: Selection-mutation differential equations: long-time behavior of measure-valued solutions. J. Differ. Equ. 261, 1472–1505 (2016)CrossRefMATHGoogle Scholar
  2. 2.
    Ackleh, A.S., Colombo, R.M., Hille, S.C., Muntean, A.: (guest editors) Modeling with measures. Math. Biosci. Eng. 12 (2015), special issueGoogle Scholar
  3. 3.
    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin, Heidelberg 1999, 2006Google Scholar
  4. 4.
    Aliprantis, C.D., Tourky, R.: Cones and Duality. American Mathematical Society, Providence (2007)CrossRefMATHGoogle Scholar
  5. 5.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, ETH Lecture Notes in Mathematics (2005)Google Scholar
  6. 6.
    Bauer, H.: Probability Theory and Elements of Measure Theory, 2nd edn. Academic Press, London (1981)MATHGoogle Scholar
  7. 7.
    Bogachev, V.I.: Measure Theory II. Springer, Berlin (2007)CrossRefMATHGoogle Scholar
  8. 8.
    Bonsall, F.F.: Linear operators in complete positive cones. Proc. Lond. Math. Soc. 8, 53–75 (1958)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Carrillo, J.A., Colombo, R.M., Gwiazda, P., Ulikowska, A.: Structured populations, cell growth and measure valued balance laws. J. Differ. Equ. 252, 3245–3277 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Carillo, J.A., Gwiazda, P., Ulikowska, A.: Splitting-particle methods for structured population models: convergence and applications. Math. Models Methods Appl. Sci. 24, 2171–2197 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cleveland, J.: Basic stage structure measure valued evolutionary game model. Math. Biosci. Eng. 12, 291–310 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Davies, R.O.: A non-Prokhorov space. Bull. Lond. Math. Soc. 3, 341–342 (1971)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Deimling, K.D.: Nonlinear Functional Analysis. Springer, Berlin (1985)CrossRefMATHGoogle Scholar
  14. 14.
    Diekmann, O., Gyllenberg, M., Metz, J.A.J., Thieme, H.R.: The ’cumulative’ formulation of (physiologically) structured population models. In: Clément, P., Lumer, G. (eds.) Evolution Equations, Control Theory, and Biomathematics, pp. 145–154. Marcel Dekker, New York (1994)Google Scholar
  15. 15.
    Dudley, R.M.: Convergence of Baire measures. Stud. Math. 27, 251–268 (1966). Correction to “Convergence of Baire measures”. Stud. Math. 51, 275 (1974)Google Scholar
  16. 16.
    Dudley, R.M.: Real Analysis and Probability, 2nd edn. Cambridge University Press, Cambridge (2002)CrossRefMATHGoogle Scholar
  17. 17.
    Dugundji, J.: Topology. Allyn and Bacon, Boston (1966)MATHGoogle Scholar
  18. 18.
    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)MATHGoogle Scholar
  19. 19.
    Evers, J.H.M., Hille, S.C., Muntean, A.: Mild solutions to a measure-valued mass evolution problem with flux boundary conditions. J. Differ. Equ. 259, 1068–1097 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Federer, H.: Geometric Measure Theory. Springer, New York (1969)MATHGoogle Scholar
  21. 21.
    Federer, H.: Colloquium lectures on geometric measure theory. Bull. Am. Math. Soc. 84, 291–338 (1978)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Fortet, R., Mourier, E.: Convergence de la répartition empirique vers la répartition théorique. Ann. Sci. Ecole Norm. Sup. 70, 266–285 (1953)MATHGoogle Scholar
  23. 23.
    Gwiazda, P., Jablonski, J., Marciniak-Czochra, A., Ulikowska, A.: Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded lipschitz distance. Numer. Methods Partial Differ. Equ. 30, 1797–1820 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Gwiazda, P., Lorenz, T., Marciniak-Czochra, A.: A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients. J. Differ. Equ. 248, 2703–2735 (2010)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Gwiazda, P., Kropielnicka, K., Marciniak-Czochra, A.: The escalator boxcar train method for a system of age-structured equations. Netw. Heterog. Media 11, 123–143 (2016)Google Scholar
  26. 26.
    Gwiazda, P., Marciniak-Czochra, A.: Structured population equations in metric spaces. J. Hyperbolic Differ. Equ. 7, 733–773 (2010)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Gwiazda, P., Orliński, P.: A. Ulikowska, Finite range method of approximation for balance laws in measure spaces, preprintGoogle Scholar
  28. 28.
    Hadeler, K.P., Waldstätter, R., Wörz-Busekros, A.: Models for pair formation in bisexual populations. J. Math. Biol. 26, 635–649 (1988)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Hadeler, K.P.: Pair formation in age-structured populations. Acta Appl. Math. 14, 91–102 (1989)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Hille, S.C., Worm, D.T.H.: Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures. Integral Equ. Oper. Theory 63, 351–371 (2009)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Hille, S.C., Worm, D.T.H.: Continuity properties of Markov semigroups and their restrictions to invariant \(L^1\) spaces. Semigroup Forum 79, 575–600 (2009)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Iannelli, M., Martcheva, M., Milner, F.A.: Gender-Structured Population Models: Mathematical Methods, Numerics, and Simulations. SIAM, Philadelphia (2005)CrossRefMATHGoogle Scholar
  33. 33.
    Jablonski, J., Marciniak-Czochra, A.: Efficient algorithms computing distances between Radon measures on R. arXiv:1304.3501, preprint
  34. 34.
    Jin, W.: Persistence of Discrete Dynamical Systems in Infinite Dimensional State Spaces, Dissertation, Arizona State University, May 2014Google Scholar
  35. 35.
    Jin, W., Smith, H.L., Thieme, H.R.: Persistence and critical domain size for diffusing populations with two sexes and short reproductive season. J. Dyn. Differ. Equ. doi: 10.1007/s10884-015-9434-1 (to appear)
  36. 36.
    Jin, W., Thieme, H.R.: Persistence and extinction of diffusing populations with two sexes and short reproductive season. Discrete Contin. Dyn. Syst. B 19, 3209–3218 (2014)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Jin, W., Thieme, H.R.: An extinction/persistence threshold for sexually reproducing populations: the cone spectral radius. Discrete Contin. Dyn. Syst. B 21, 447–470 (2016)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Krasnosel’skij, M.A.: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964)MATHGoogle Scholar
  39. 39.
    Krasnosel’skij, M.A., Lifshits, JeA, Sobolev, A.V.: Positive Linear Systems: The Method of Positive Operators. Heldermann Verlag, Berlin (1989)Google Scholar
  40. 40.
    Krause, U.: Positive Dynamical Systems in Discrete Time. Theory, Models and Applications. De Gruyter, Berlin (2015)CrossRefMATHGoogle Scholar
  41. 41.
    Krein, M.G.: Sur les opérations linéaires transformant un certain ensemble conique en lui-même. C.R. (Doklady) Acad. Sci. U.R.S.S. (N.S.) 3(2), 749–752 (1939)MATHGoogle Scholar
  42. 42.
    Krein, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space (Russian). Uspehi Mat. Nauk (N.S.)3 (1948), 3–95. English Translation, AMS Translation 1950 (1950), No. 26Google Scholar
  43. 43.
    Lang, S.: Analysis II. Addison Wesley, Reading (1969)MATHGoogle Scholar
  44. 44.
    Lasota, A., Myjak, J., Szarek, T.: Markov operators with a unique invariant measure. J. Math. Anal. Appl. 276, 343–356 (2002)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Lemmens, B., Nussbaum, R.D.: Nonlinear Perron-Frobenius Theory. Cambridge University Press, Cambridge (2012)CrossRefMATHGoogle Scholar
  46. 46.
    Mallet-Paret, J., Nussbaum, R.D.: Eigenvalues for a class of homogeneous cone maps arising from max-plus operators. Discrete Contin. Dyn. Syst. A 8, 519–562 (2002)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Mallet-Paret, J., Nussbaum, R.D.: Generalizing the Krein–Rutman theorem, measures of noncompactness and the fixed point index. J. Fixed Point Theory Appl. 7, 103–143 (2010)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    McDonald, J.N., Weiss, N.A.: A Course in Real Analysis. Academic Press, San Diego (1999)MATHGoogle Scholar
  49. 49.
    Neunzert, H.: An introduction to the nonlinear Boltzmann-Vlasov equation. In: Kinetic Theories and the Boltzmann Equation (Montecatini, 1981), Lecture Notes in Math., vol. 1048. Springer, Berlin, pp. 60–110 (1984)Google Scholar
  50. 50.
    Nussbaum, R.D.: Eigenvectors of nonlinear positive operators and the linear Krein–Rutman theorem. In: Fadell, E., Fournier, G. (eds.) Fixed Point Theory, pp. 309–331. Springer, Berlin (1981)CrossRefGoogle Scholar
  51. 51.
    Nussbaum, R.D.: Hilbert’s projective metric and iterated nonlinear maps. Mem. AMS 75, Number 391, Am. Math. Soc., Providence (1988)Google Scholar
  52. 52.
    Nussbaum, R.D.: Eigenvectors of order-preserving linear operators. J. Lond. Math. Soc. 2, 480–496 (1998)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Rudin, W.: Real and Complex Analysis, 2nd edn. McGraw-Hill, New York (1966)MATHGoogle Scholar
  54. 54.
    Schaefer, H.H.: Halbgeordnete lokalkonvexe Vektorräume. II. Math. Ann. 138, 259–286 (1959)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Schaefer, H.H.: Topological Vector Spaces. Macmillan, New York (1966)MATHGoogle Scholar
  56. 56.
    Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. American Mathematical Society, Providence (2011)MATHGoogle Scholar
  57. 57.
    Taira, K.: Semigroups, Boundary Value Problems and Markov Processes. Springer, Berlin (2004)CrossRefMATHGoogle Scholar
  58. 58.
    Thieme, H.R.: Well-posedness of physiologically structured population models for Daphnia magna (How biological concepts can benefit by abstract mathematical analysis). J. Math. Biol. 26, 299–317 (1988)MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Thieme, H.R.: Eigenvectors and eigenfunctionals of homogeneous order-preserving maps. arXiv:1302.3905v1 [math.FA] (2013)
  60. 60.
    Thieme, H.R.: Spectral radii and Collatz–Wielandt numbers for homogeneous order-preserving maps and the monotone companion norm. In: de Jeu, M., de Pagter, B., van Gaans, O., Veraar, M. (eds.) Ordered Structures and Applications, Positivity VII (Zaanen Centennial Conference). Birkhäuser, Springer International Publishing, Switzerland, pp. 415–467Google Scholar
  61. 61.
    Ulikowska, A.: Structured Population Models in Metric Spaces. Dissertation, Warsaw University (2013)Google Scholar
  62. 62.
    van den Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer, New York (1996)CrossRefMATHGoogle Scholar
  63. 63.
    Worm, D.: Semigroups on Spaces of Measures. Dissertation, Leiden (2010)Google Scholar
  64. 64.
    Yosida, K.: Functional Analysis, 2nd edn. Springer, Berlin, Heidelberg (1965-1968)Google Scholar
  65. 65.
    Zhao, X.-Q.: Dynamical Systems in Population Biology. Springer, New York (2003)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Piotr Gwiazda
    • 1
  • Anna Marciniak-Czochra
    • 2
  • Horst R. Thieme
    • 3
  1. 1.Institute of Applied Mathematics and MechanicsUniversity of WarsawWarsawPoland
  2. 2.Institute of Applied MathematicsUniversity of HeidelbergHeidelbergGermany
  3. 3.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA

Personalised recommendations