INITIAL PROBLEM FOR A THIRD ORDER NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF CONVOLUTION TYPE

Abstract

The article obtains two-sided a priori estimates for the solution of a homogeneous third-order Volterra integro-differential equation with power-law nonlinearity and a difference kernel. It is shown that the lower a priori estimate, which plays the role of a weight function when constructing a metric in the cone of the space of continuous functions, is unimprovable. Using these estimates, using the method of weight metrics (analogous to A. Bielecki’s method), a global theorem on the existence, uniqueness and method of finding a nontrivial solution to the initial problem for the specified integro-differential equation in the class of non-negative continuous functions on the positive half-axis is proved. It is shown that the solution can be found by the method of successive approximations and an estimate of the rate of their convergence to the exact solution is obtained. Examples are given to illustrate the results obtained.