Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces
Volume 62 / 1995
Annales Polonici Mathematici 62 (1995), 13-21
DOI: 10.4064/ap-62-1-13-21
Abstract
We investigate the structure of the set of solutions of the Cauchy problem x' = f(t,x), x(0) = x₀ in Banach spaces. If f satisfies a compactness condition expressed in terms of measures of weak noncompactness, and f is Pettis-integrable, then the set of pseudo-solutions of this problem is a continuum in $C_{w}(I,E)$, the space of all continuous functions from I to E endowed with the weak topology. Under some additional assumptions these solutions are, in fact, weak solutions or strong Carathéodory solutions, so we also obtain Kneser-type theorems for these classes of solutions.
