Analysis of a nonlinear delay integro-differential equation involving weakly singular kernels

The purpose of this work is to establish analytical and numerical treatments for a nonlinear delay integro-differential equation involving weakly singular kernel. Firstly, we discuss the questions of the solution existence and uniqueness, after that we construct a numerical approach base essentially on approximating the integral operators using the product integration method then solving the obtained nonlinear algebraic system by Picard’s iterations. The convergence of the approximation process is proved and the compatibility of the proposed method with the analytical conditions is shown via a numerical example.