Abstract
In this paper we give a generalization to recent results by using weak and strong measures of noncompactness. For
f:[0,
T]×
E→
E with
E is a Banach space we prove that, under suitable assumptions, the Cauchy problem
(P)
x
̇
(t)=f(t,x(t)),
t∈[0,T],
x(0)=x
0,
has at least one weak solution furthermore, with certain conditions, the Cauchy problem (
P) has a solution. Next under a generalization of the compactness assumptions, we show that (
P) has a solution too.