Initial-value problem for an integro-differential equation with difference kernels and an inhomogeneity in the linear part

A global theorem on the existence and uniqueness of a nonnegative solution of the initial-value problem for an integro-differential equation with difference kernels, power nonlinearity, and inhomogeneity in the linear part is proved by the method of weight metrics in the cone of the space of continuous functions. It is shown that the solution can be found by the method of successive approximations of the Picard type. An estimate of the rate of their convergence is obtained.