EMIS ELibM Electronic Journals Zeitschrift für Analysis und ihre Anwendungen
Vol. 18, No. 3, pp. 569-584 (1999)

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Compactness and Existence Results for Ordinary Differential Equations in Banach Spaces

J. Appell, M. Väth and A. Vignoli

J. Appell: Dept. Math. Univ., Am Hubland, D-97074 Würzburg; M. Väth: Dept. Math. Univ., Am Hubland, D-97074 Würzburg; A. Vignoli: Univ. of Rome ``Tor Vergata'', Dept. Math., Via della Ricerca Sci., I-00133 Roma

Abstract: We prove that the Picard-Lindelöf operator $$ Hx(t) = \int_{t_0}^t f(s,x(s))\,ds $$ 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

Classification (MSC2000): 34A12, 34G20, 47H30, 45N10, 45P05, 46G10

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Electronic fulltext finalized on: 7 Aug 2001. This page was last modified: 9 Nov 2001.

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