JournalszaaVol. 18, No. 3pp. 569–584

Compactness and Existence Results for Ordinary Differential Equations in Banach Spaces

  • Jürgen Appell

    Universität Würzburg, Germany
  • Martin Väth

    Czech Academy of Sciences, Prague, Czech Republic
  • A. Vignoli

    Università di Roma 'Tor Vergata', Italy
Compactness and Existence Results for Ordinary Differential Equations in Banach Spaces cover
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Abstract

We prove that the Picard-Lindelöf operator

Hx(t)=t0tf(s,x(s))dsHx(t) = \int^t_{t_0} f(s, x(s)) ds

with a vector function ff is continuous and compact (condensing) in CC, if ff satisfies only a mild boundedness condition, and if f(s,)f(s,\cdot) is continuous and compact (resp. condensing). This generalizes recent results of the second author and immediately leads to existence theorems for local weak solutions of the initial value problem for ordinary differential equations in Banach spaces.

Cite this article

Jürgen Appell, Martin Väth, A. Vignoli, Compactness and Existence Results for Ordinary Differential Equations in Banach Spaces. Z. Anal. Anwend. 18 (1999), no. 3, pp. 569–584

DOI 10.4171/ZAA/899