On a Theorem of Mierczyński

Gerd Herzog

Colloquium Mathematicae (1998)

  • Volume: 76, Issue: 1, page 19-29
  • ISSN: 0010-1354

Abstract

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We prove that the initial value problem x’(t) = f(t,x(t)), x ( 0 ) = x 1 is uniquely solvable in certain ordered Banach spaces if f is quasimonotone increasing with respect to x and f satisfies a one-sided Lipschitz condition with respect to a certain convex functional.

How to cite

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Herzog, Gerd. "On a Theorem of Mierczyński." Colloquium Mathematicae 76.1 (1998): 19-29. <http://eudml.org/doc/210548>.

@article{Herzog1998,
abstract = {We prove that the initial value problem x’(t) = f(t,x(t)), $x(0) = x_1$ is uniquely solvable in certain ordered Banach spaces if f is quasimonotone increasing with respect to x and f satisfies a one-sided Lipschitz condition with respect to a certain convex functional.},
author = {Herzog, Gerd},
journal = {Colloquium Mathematicae},
keywords = {initial value problems; unique solution; continuous dependence; ordered Banach space},
language = {eng},
number = {1},
pages = {19-29},
title = {On a Theorem of Mierczyński},
url = {http://eudml.org/doc/210548},
volume = {76},
year = {1998},
}

TY - JOUR
AU - Herzog, Gerd
TI - On a Theorem of Mierczyński
JO - Colloquium Mathematicae
PY - 1998
VL - 76
IS - 1
SP - 19
EP - 29
AB - We prove that the initial value problem x’(t) = f(t,x(t)), $x(0) = x_1$ is uniquely solvable in certain ordered Banach spaces if f is quasimonotone increasing with respect to x and f satisfies a one-sided Lipschitz condition with respect to a certain convex functional.
LA - eng
KW - initial value problems; unique solution; continuous dependence; ordered Banach space
UR - http://eudml.org/doc/210548
ER -

References

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  1. [1] A. Chaljub-Simon, R. Lemmert, S. Schmidt and P. Volkmann, Gewöhnliche Differentialgleichungen mit quasimonoton wachsenden rechten Seiten in geordneten Banachräumen, in: General Inequalities 6 (Oberwolfach, 1990), Internat. Ser. Numer. Math. 103, Birkhäuser, Basel, 1992, 307-320. Zbl0763.34048
  2. [2] K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math. 296, Springer, Berlin, 1977. 
  3. [3] G. Herzog, An existence and uniqueness theorem for ordinary differential equations in ordered Banach spaces, Demonstratio Math., to appear. Zbl0911.34055
  4. [4] G. Herzog, On ordinary differential equations with quasimonotone increasing right hand side, Arch. Math. (Basel), to appear. Zbl0896.34056
  5. [5] R. Lemmert, Existenzsätze für gewöhnliche Differentialgleichungen in geordneten Banachräumen, Funkcial. Ekvac. 32 (1989), 243-249. Zbl0721.34073
  6. [6] R. Lemmert, R. M. Redheffer and P. Volkmann, Ein Existenzsatz für gewöhnliche Differentialgleichungen in geordneten Banachräumen, in: General Inequalities 5 (Oberwolfach, 1986), Internat. Ser. Numer. Math. 80, Birkhäuser, Basel, 1987, 381-390. Zbl0625.34070
  7. [7] R. Lemmert, S. Schmidt and P. Volkmann, Ein Existenzsatz für gewöhnliche Differentialgleichungen mit quasimonoton wachsender rechter Seite, Math. Nachr. 153 (1991), 349-352. 
  8. [8] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Krieger, 1987. 
  9. [9] J. Mierczyński, Strictly cooperative systems with a first integral, SIAM J. Math. Anal. 18 (1987), 642-646. Zbl0657.34033
  10. [10] J. Mierczyński, Uniqueness for a class of cooperative systems of ordinary differential equations, Colloq. Math. 67 (1994), 21-23. Zbl0831.34001
  11. [11] J. Mierczyński, Uniqueness for quasimonotone systems with strongly monotone first integral, in: Proc. Second World Congress of Nonlinear Analysts (WCNA-96), Athens, 1996, to appear. Zbl0896.34001
  12. [12] P. Volkmann, Gewöhnliche Differentialungleichungen mit quasimonoton wachsenden Funktionen in topologischen Vektorräumen, Math. Z. 127 (1972), 157-164. Zbl0226.34058
  13. [13] P. Volkmann, Cinq cours sur les équations différentielles dans les espaces de Banach, in: Topological Methods in Differential Equations and Inclusions (Montréal, 1994), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 472, Kluwer, Dordrecht, 1995, 501-520. 

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