The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (1085 KB) | Metadata | Table of Contents | ZAA summary
Volume 18, Issue 3, 1999, pp. 569–584
DOI: 10.4171/ZAA/899

Published online: 1999-09-30

Compactness and Existence Results for Ordinary Differential Equations in Banach Spaces

Jürgen Appell[1], Martin Väth[2] and A. Vignoli[3]

(1) Universität Würzburg, Germany
(2) Czech Academy of Sciences, Prague, Czech Republic
(3) Università di Roma 'Tor Vergata', Italy

We prove that the Picard-Lindelöf operator $$Hx(t) = \int^t_{t_0} f(s, x(s)) ds$$ with a vector function $f$ is continuous and compact (condensing) in $C$, if $f$ satisfies only a mild boundedness condition, and if $f(s,\cdot)$ 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.

Keywords: Ordinary differential equations in Banach spaces, nonlinear Volterra integral operators, Picard-Lindelöf operators, compactness, condensing operators, measures of non-compactness

Appell Jürgen, Väth Martin, Vignoli A.: Compactness and Existence Results for Ordinary Differential Equations in Banach Spaces. Z. Anal. Anwend. 18 (1999), 569-584. doi: 10.4171/ZAA/899