In this paper, under certain conditions, we prove the existence of at least one solution of Volterra-Hammerstein integral equation of the second kind with singular kernel in the space   using Schauder fixed point theorem. Then, using a numerical method, we have a system of Hammerstein integral equations. Therefore, some important theorems, using Schauder fixed point theorem, must be discussed. Moreover, Toeplitz matrix method is used to obtain a nonlinear algebraic system, where the solution (at least one solution) can be discussed. The estimate error of the numerical method is established and computed.

Volterra-Hammerstein integral equation (V-HIE), nonlinear algebraic system (NAS), singular kernel, Toeplitz matrix method.