Volterra type integro-differential equation with a sum-difference kernel and power nonlinearity

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