ArticleOriginal scientific text

Title

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

Authors 1

Affiliations

  1. Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland

Abstract

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.

Keywords

Volterra integral equation in a Banach space, δ-sets

Bibliography

  1. A. Alexiewicz, Functional Analysis, PWN, Warszawa, 1969 (in Polish).
  2. N. Aronszajn, Le correspondant topologique de l'unicité dans la théorie des équations différentielles, Ann. of Math. 43 (1942), 730-738.
  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.
  4. K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math. 596, Springer, Berlin, 1977.
  5. K. Goebel, Thickness of sets in metric spaces and applications in fixed point theory, habilitation thesis, Lublin, 1970 (in Polish).
  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.
  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. S. Szufla, On the structure of solution sets of differential and integral equations in Banach spaces, Ann. Polon. Math. 34 (1977), 165-177.
  9. G. Vidossich, On the structure of the set of solutions of nonlinear equations, J. Math. Anal. Appl. 34 (1971), 602-617.
Pages:
33-39
Main language of publication
English
Received
1991-09-06
Accepted
1992-04-22
Published
1994
Exact and natural sciences