On the set of solutions of the Cauchy problem in Banach spaces (Q1337777)

This paper contains some Kneser type theorems for an abstract Cauchy problem \(x'= f(t,x)\), \(x(0)= x_ 0\) in an infinite-dimensional Banach space \(E\). A function \(f\) is assumed to satisfy Carathéodory-type conditions with respect to the weak topology on \(E\). However, some compactness conditions expressed in terms of a measure of weak noncompactness is assumed. Under the assumptions mentioned above a nonemptiness, compactness and connectedness of the set of all pseudo- solutions for the Cauchy problem is proved. Topological properties of this set are considered in the space \(C(J,E)\) with its weak topology.