Initial Value Problem for a Third-Order Nonlinear Integro-Differential Equation of Convolution Type

Abstract

In this paper, we obtain two-sided a priori estimates for the solution of a homogeneous third-order Volterra integro-differential equation with a power-law nonlinearity and 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 sharp. Using these estimates, by the weighted metric method (an analog of A. Bielecki’s method), we prove a global theorem on the existence and uniqueness of a nontrivial solution of the initial value problem for this integro-differential equation in the class of nonnegative continuous functions on the positive half-line and on the method for finding this solution. It is shown that the solution can be found by the successive approximation method, and an estimate of the rate of convergence of the approximations to the exact solution is obtained. Examples are given to illustrate the results obtained.