We prove the existence of solutions, in separable Banach spaces, for the following differential inclusion: \begin{align*} eft\{\begin{array}{ll} \dot{x}(t) ı F(t,T(t)x),\quad &\mbox{a.e. on }[0,au]; x(s)=ǎrphi(s),\quad &\forall sı [-a,0]; x(t) ı C(t),\quad &\forall tı [0,au]; \end{array} \right. \end{align*} We consider weaker hypotheses on the constraint.