On solubility of hypersingular integral equations

Background. Hypersingular integral equations are an actively developing field of mathematical physics. It is associated with numerous applications of hypersingular integral equations in aerodynamics, electrodynamics, quantum physics, geophysics. Besides direct applications in physics and technology, hypersingular integral equations occur when solving boundary problems of mathematical physics. Recently there has been published a series of works devoted to approximate methods of solving hypersingular integral equations of first and second kind on closed and open integration contours. The interest to these methods is associated with direct applications of hypersingular integral equations in aerodynamics and electrodynamics. At the same time there is no general theory of hypersingular integral equations - there are no confirmations on existence and uniqueness of solutions of hypersingular integral equations. The present article describes a number of assertions on solubility of hypersingular integral equations. The presence of such assertions allows to use hypersingular integral equations more efficiently in many applications. Materials and methods. During the study the author used methods of functional analysis, singular integral equations theory and generalized Riemann boundary problems. The article considers linear one-dimensional hypersingular integral equations on closed integration contours. Results. The researcher obtained general confirmations on existence and uniqueness of hypersingular integral equations, set on closed integration contours. Conclusions. The author obtained general confirmation on existence of solutions of hypersingular integral equations. When solving applied problems, these confirmations allow to set the problem of finding all solutions of a problem under consideration. The obtained results may be used for solving problems of aerodynamics, electrodynamics, when solving equations of mathematical physics by the method of boundary integral equations.