Solvability and construction of solutions of integral equations

Abstract

A class of integral equations with respect to one variable function as well as to multivariable function that are solvable for any right hand side of an equation has been singled out. A necessary and sufficient condition for existence of a solution has been obtained for the class of integral equations and the general form of their exact solutions has been found. Necessary and sufficient conditions for existence of solutions to the mentioned equations with a given right hand side are obtained by reducing them to solving an extremal problem. An algorithm for solving the extremal problem by constructing a minimizing sequence has been developed and a convergence rate estimation has been obtained. A solvability criterion as a requirement on infimum of functional has been formulated. A necessary and sufficient condition for solvability of an integral equation with parameter has been obtained and its general solution has been found.

Keywords

• integral equation
• general solution
• existence of a solution
• necessary and sufficient condition
• solvability criterion
• extremal problem
• minimizing sequence