Well-posedness of a non-local abstract Cauchy problem with singular integral

In this work, the abstract Cauchy problem for an initial value system with singular integral is considered. The system is of closed form of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, we prove the existence and uniqueness of classical solutions to the evolution system under assumptions on the boundedness and smoothness of data. Furthermore, we connect by an isomorphism the solution of the evolution system and a class of integral-differential equations with singular convolution kernels and extend our results to the corresponding problem. It is revealed that our findings also improve the understanding of the open problem on the well-posedness of the stationary Wigner equation with inflow boundary conditions.