Vector-valued measures of noncompactness and the Cauchy problem with delay in a scale of Banach spaces

In this paper, by developing a new method we advance to prove the existence results for the Cauchy problem with delay d t u = f ( t , u ( t ) , u ( h ( t ) ) ) , t ∈ ( 0 , T ) , u ( 0 ) = u 0 , in a scale of Banach spaces. The arguments are based upon Darbo–Sadovskii fixed point theorem for condensing operators on Fréchet spaces and the technique of vector-valued measure of noncompactness.