On a Theorem of Mierczyński

Gerd Herzog

Colloquium Mathematicae (1998)

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

Abstract

top
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

top

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

top
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.