Estimates of the solutions of two-point boundary-value problems for systems of first-order integrodifferential equations and the method of lines

Abstract

One proves theorems on the estimates of the solutions of the systems of first-order integrodifferential equations

with the boundary conditions

On the basis of these theorems, one suggests a method for estimating the norms of integrodifferential equations by the method of the lines for the solutions of the periodic boundary-value problems for second-order integrodifferential equations of parabolic type. On the basis of the established theorem, on the solvability and on the estimate of the solution of the nonlinear equation

$$Tx + F\left( x \right) = 0$$

in a Banach space X, where T is a linear unbounded operator, one investigates the convergence of the method of lines for solving the periodic boundary-value problem for a second-order nonlinear integrodifferential equation of parabolic type.