Existence of solutions for Volterra singular integral equations in the class of differentiable functions

Abstract

In this paper, our purpose is to obtain the general solutions of several kinds of Volterra singular integral equations (VSIEs) in the class of differentiable functions. By constructing some operators and using the properties of integral transforms and conformal mappings, we transform VSIEs in the class of differentiable functions into the Riemann-Hilbert problems with discontinuity on a circle. By means of the principle of analytic continuation and Sokhotski-Plemelj formula, we obtain solutions of Riemann-Hilbert problems in the case of non-normal type, and further discuss the asymptotic properties of the solutions at the nodes.