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

Abstract

In this paper, by developing a new method we advance to prove the existence results for the Cauchy problem with delay $$d_tu=f(t,u(t),u(h(t))),\;t\in (0,T),\;u(0)=u_0$$, in a scale of Banach spaces. The arguments are based upon Darbo–Sadovskii fixed point theorem for condensing operators on Frechet spaces and the technique of vector-valued measure of noncompactness.