Compactness of nonlinear integral operators with discontinuous and with singular kernels

This paper establishes compactness of nonlinear integral operators in the space continuous functions. One result deals with whose kernel can have jumps across a finite number curves, which typically arise from study ordinary differential equations boundary conditions local or nonlocal type. Several other results deal kernels singularity, fractional equations. We motivate these by discussing some initial value problems for Caputo and Riemann-Liouville prove compact embedding theorem integrals order to give new treatment singular case.