Zeitschrift für Analysis und ihre Anwendungen Vol. 18, No. 3, pp. 569584 (1999) 

Compactness and Existence Results for Ordinary Differential Equations in Banach SpacesJ. Appell, M. Väth and A. VignoliJ. Appell: Dept. Math. Univ., Am Hubland, D97074 Würzburg; M. Väth: Dept. Math. Univ., Am Hubland, D97074 Würzburg; A. Vignoli: Univ. of Rome ``Tor Vergata'', Dept. Math., Via della Ricerca Sci., I00133 RomaAbstract: We prove that the PicardLindelöf 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, PicardLindelöf operators, compactness, condensing operators, measures of noncompactness Classification (MSC2000): 34A12, 34G20, 47H30, 45N10, 45P05, 46G10 Full text of the article:
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