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Abstract Let L1(I) be the space of Lebesgue integrable functions defined on the interval I = [0; T]. Let E be a Banach space with norm k : k and its dual E*. Denote by C[I;E] the Banach space of strongly continuous functions x : I ! E with sup-norm. Denote by E = (E; !) = (E; (E;E*)) the space E with its weak topology, C[I;E!] denote the space of weakly continuous functions on I endowed with the topology of weak uniform convergence. Let f(:; :) : I x E ! E. .Consider the Cauchy problem |