On the structure of the set of solutions of a Volterra integral equation in a Banach space

Krzysztof Czarnowski

Annales Polonici Mathematici (1994)

  • Volume: 59, Issue: 1, page 33-39
  • ISSN: 0066-2216

Abstract

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The set of solutions of a Volterra equation in a Banach space with a Carathéodory kernel is proved to be an δ , in particular compact and connected. The kernel is not assumed to be uniformly continuous with respect to the unknown function and the characterization is given in terms of a B₀-space of continuous functions on a noncompact domain.

How to cite

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Krzysztof Czarnowski. "On the structure of the set of solutions of a Volterra integral equation in a Banach space." Annales Polonici Mathematici 59.1 (1994): 33-39. <http://eudml.org/doc/262486>.

@article{KrzysztofCzarnowski1994,
abstract = {The set of solutions of a Volterra equation in a Banach space with a Carathéodory kernel is proved to be an $ℛ_δ$, in particular compact and connected. The kernel is not assumed to be uniformly continuous with respect to the unknown function and the characterization is given in terms of a B₀-space of continuous functions on a noncompact domain.},
author = {Krzysztof Czarnowski},
journal = {Annales Polonici Mathematici},
keywords = {Volterra integral equation in a Banach space; $ℛ_δ$-sets; solution set; nonlinear Volterra integral equation of the second kind; abstract Banach space; Carathéodory conditions},
language = {eng},
number = {1},
pages = {33-39},
title = {On the structure of the set of solutions of a Volterra integral equation in a Banach space},
url = {http://eudml.org/doc/262486},
volume = {59},
year = {1994},
}

TY - JOUR
AU - Krzysztof Czarnowski
TI - On the structure of the set of solutions of a Volterra integral equation in a Banach space
JO - Annales Polonici Mathematici
PY - 1994
VL - 59
IS - 1
SP - 33
EP - 39
AB - The set of solutions of a Volterra equation in a Banach space with a Carathéodory kernel is proved to be an $ℛ_δ$, in particular compact and connected. The kernel is not assumed to be uniformly continuous with respect to the unknown function and the characterization is given in terms of a B₀-space of continuous functions on a noncompact domain.
LA - eng
KW - Volterra integral equation in a Banach space; $ℛ_δ$-sets; solution set; nonlinear Volterra integral equation of the second kind; abstract Banach space; Carathéodory conditions
UR - http://eudml.org/doc/262486
ER -

References

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  1. [1] A. Alexiewicz, Functional Analysis, PWN, Warszawa, 1969 (in Polish). 
  2. [2] N. Aronszajn, Le correspondant topologique de l'unicité dans la théorie des équations différentielles, Ann. of Math. 43 (1942), 730-738. Zbl0061.17106
  3. [3] K. Czarnowski and T. Pruszko, On the structure of fixed point sets of compact maps in B₀ spaces with applications to integral and differential equations in unbounded domain, J. Math. Anal. Appl. 154 (1991), 151-163. Zbl0729.47054
  4. [4] K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math. 596, Springer, Berlin, 1977. 
  5. [5] K. Goebel, Thickness of sets in metric spaces and applications in fixed point theory, habilitation thesis, Lublin, 1970 (in Polish). 
  6. [6] H. P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal. 7 (1983), 1351-1371. Zbl0528.47046
  7. [7] J. M. Lasry et R. Robert, Analyse non linéaire multivoque, Centre de Recherche de Math. de la Décision, No. 7611, Université de Paris-Dauphine. 
  8. [8] S. Szufla, On the structure of solution sets of differential and integral equations in Banach spaces, Ann. Polon. Math. 34 (1977), 165-177. Zbl0384.34038
  9. [9] G. Vidossich, On the structure of the set of solutions of nonlinear equations, J. Math. Anal. Appl. 34 (1971), 602-617. 

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