Volterra integro-differential equation of arbitrary order with power nonlinearity

In the cone of the space of continuous functions, the method of weight metrics (analogous to Bielecki's method) is used to prove a global theorem on the existence, uniqueness, and method of finding a nontrivial solution to the initial problem for a homogeneous $n$-order integro-differential equation with a difference kernel and power nonlinearity. 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 to the solution in terms of the weight metric. The study is based on the reduction of the initial problem to the equivalent nonlinear Volterra integral equation. Exact lower and upper a priori estimates for the solution are obtained, on the basis of which a complete weighted metric space is constructed that is invariant with respect to the nonlinear operator generated by this Volterra integral equation. In contrast to the linear case, it has been established that, in addition to the trivial solution, the non-linear homogeneous Volterra integral equation can also have a non-trivial solution. An analysis of the results obtained shows that with an increase in the order of an integro-differential equation with a power nonlinearity, the exponent decreases. Examples are given to illustrate the results obtained.