Weak and strong solutions for differential equations in Banach spaces

n 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 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.