On integral equations of Fredholm kind in Bohr space of almost periodic functions

Abstract

In the present work we consider a question on extending the notion of the Fredholm integral equation or second kind integral equation, which allows one to consider the issue on existence of solutions in the space of almost periodic functions. Almost periodic functions are defined on the entire line. This is why it seems difficult to describe them by some characteristics on finite intervals.The Fredholm equations are known to be closely related with first order differential equations. In some particular cases there were posed the questions on finding the solutions in various classes of almost periodic functions. In some known cases there are no solutions in the Bohr class for such equations with almost periodic coefficients.There are known examples of almost periodic functions (in the Besicovitch sense), which can not be solutions for a rather wide class of differential equations. It is natural to expect that in the general case the integral equations are also not solvable in Bohr class of almost-periodic functions. This is why a more specific approach is needed for the problem in the space of almost-periodic functions.