Evaluation maps, restriction maps, and compactness

Elizabeth Bator; Paul Lewis; James Ochoa

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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...