In this work, we use the De Blasi measure of noncompactness and Darbo fixed point Theorem to study the existence of solutions for an initial value problem of a delay implicit functional integro-differential equation. The sufficient condition for the uniqueness of the solution will be given. The continuous dependence of the unique solution on some data will be studied.
The study of implicit differential and integral equations has received much attention over the last 30 years or so. For papers studying such kind of problems (see \cite{15,16,17,18})
and the references therein.
For the theoretical results concerning the existence of solutions, in the classes of continuous or integrable functions, you can see Bana´s [18–21]. Each of these monographs contains some existence results, and the main objective is to present a technique to obtain some results concerning various integral equations.
Here we are concerning with the initial value problem of the delay implicit functional integro-differential equation

dx/dt=f(t,dx/dt,∫_0^ϕ(t)▒‍ g(s,x(s))ds),a.e t∈(0,1] (1)

with the initial data
x(0)=x_0,