Necessary and Sufficient Conditions for the Solvability of Linear Boundary-Value Problems for the Fredholm Integrodifferential Equations

Abstract

We propose a method for the investigation and solution of linear boundary-value problems for the Fredholm integrodifferential equations based on the partition of the interval and introduction of additional parameters. Every partition of the interval is associated with a homogeneous Fredholm integral equation of the second kind. The definition of regular partitions is presented. It is shown that the set of regular partitions is nonempty. A criterion for the solvability of the analyzed problem is established and an algorithm for finding its solutions is constructed. Integrodifferential equations are often encountered in applications as mathematical models of various processes in natural sciences. In the monograph [1], their role in the study of hereditary processes is indicated and a survey of the papers devoted to the boundary-value problems for the Volterra and Fredholm integrodifferential equations is presented. The solvability of various problems for the Fredholm integrodifferential equations and approximate methods for finding their solutions are studied by many authors [2–11]. Linear boundary-value problems for Fredholm integrodifferential equations with degenerate kernels are considered in the monograph [12]. Necessary and sufficient conditions for the solvability of boundary-value problems are established and the algorithms for finding the solutions of these problems are constructed. In the present paper, we consider a linear two-point boundary-value problem