Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces

Mieczysław Cichoń; Ireneusz Kubiaczyk

Annales Polonici Mathematici (1995)

  • Volume: 62, Issue: 1, page 13-21
  • ISSN: 0066-2216

Abstract

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We investigate the structure of the set of solutions of the Cauchy problem x’ = f(t,x), x(0) = x₀ in Banach spaces. If f satisfies a compactness condition expressed in terms of measures of weak noncompactness, and f is Pettis-integrable, then the set of pseudo-solutions of this problem is a continuum in C w ( I , E ) , the space of all continuous functions from I to E endowed with the weak topology. Under some additional assumptions these solutions are, in fact, weak solutions or strong Carathéodory solutions, so we also obtain Kneser-type theorems for these classes of solutions.

How to cite

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Mieczysław Cichoń, and Ireneusz Kubiaczyk. "Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces." Annales Polonici Mathematici 62.1 (1995): 13-21. <http://eudml.org/doc/262264>.

@article{MieczysławCichoń1995,
abstract = {We investigate the structure of the set of solutions of the Cauchy problem x’ = f(t,x), x(0) = x₀ in Banach spaces. If f satisfies a compactness condition expressed in terms of measures of weak noncompactness, and f is Pettis-integrable, then the set of pseudo-solutions of this problem is a continuum in $C_\{w\}(I,E)$, the space of all continuous functions from I to E endowed with the weak topology. Under some additional assumptions these solutions are, in fact, weak solutions or strong Carathéodory solutions, so we also obtain Kneser-type theorems for these classes of solutions.},
author = {Mieczysław Cichoń, Ireneusz Kubiaczyk},
journal = {Annales Polonici Mathematici},
keywords = {set of solutions; pseudo-solutions; measures of weak noncompactness; Pettis integral; Cauchy problem; Banach space},
language = {eng},
number = {1},
pages = {13-21},
title = {Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces},
url = {http://eudml.org/doc/262264},
volume = {62},
year = {1995},
}

TY - JOUR
AU - Mieczysław Cichoń
AU - Ireneusz Kubiaczyk
TI - Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces
JO - Annales Polonici Mathematici
PY - 1995
VL - 62
IS - 1
SP - 13
EP - 21
AB - We investigate the structure of the set of solutions of the Cauchy problem x’ = f(t,x), x(0) = x₀ in Banach spaces. If f satisfies a compactness condition expressed in terms of measures of weak noncompactness, and f is Pettis-integrable, then the set of pseudo-solutions of this problem is a continuum in $C_{w}(I,E)$, the space of all continuous functions from I to E endowed with the weak topology. Under some additional assumptions these solutions are, in fact, weak solutions or strong Carathéodory solutions, so we also obtain Kneser-type theorems for these classes of solutions.
LA - eng
KW - set of solutions; pseudo-solutions; measures of weak noncompactness; Pettis integral; Cauchy problem; Banach space
UR - http://eudml.org/doc/262264
ER -

References

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