Integro-differential equation with a sum-difference kernels and power nonlinearity

Sharp a priori estimates are obtained for solutions to a nonlinear Volterra-type integro-differential equation with a sum-difference kernel in a cone of the space of functions continuous on the positive half-axis. On the basis of these estimates, the method of weighted metrics is used to prove a global theorem on the existence, uniqueness, and a method of finding a nontrivial solution of the indicated equation. It is shown that this solution can be found by a method of successive approximations of the Picard type, and an estimate is given for the rate of their convergence in terms of the weighted metric. Conditions under which only the trivial solution exists are indicated. Examples are given to illustrate the results.