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

In this paper, we obtain two-sided a priori estimates for the solution of a homogeneousthird-order Volterra integro-differential equation with a power-law nonlinearity and differencekernel. It is shown that the lower a priori estimate, which plays the role of a weight function whenconstructing a metric in the cone of the space of continuous functions, is sharp. Using theseestimates, by the weighted metric method (an analog of A. Bielecki’s method), we prove a globaltheorem on the existence and uniqueness of a nontrivial solution of the initial value problem forthis integro-differential equation in the class of nonnegative continuous functions on the positivehalf-line and on the method for finding this solution. It is shown that the solution can be found bythe successive approximation method, and an estimate of the rate of convergence of theapproximations to the exact solution is obtained. Examples are given to illustrate the resultsobtained.