Uniqueness and Existence of Solutions to Some Kinds of Singular Convolution Integral Equations with Cauchy Kernel via R-H Problems

In this article, we study the existence and uniqueness of solutions to some kinds of singular integral equations with Cauchy kernel and convolution kernel. In order to transform our equations into Riemann-Hilbert problems (R-H problems), we establish the relation between Fourier integral transform and Cauchy type integral, and we generalize Sokhotski-Plemelj formula. By means of the classical R-H problems and of regularity theory of Fredholm integral equations, we present the necessary and sufficient conditions of Noether solvability and the explicit solutions for such equations. Especially, we discuss the property and asymptotic behavior of solutions at nodes. This paper will be of great significance for the study of improving and developing complex analysis, integral equations, and R-H problems.