Solvability of systems of integral Volterra equations of the first kind with piecewise continuous kernels

We construct an asymptotic approximation for solutions of systems of integral Volterra equations of the first kind with piecewise continuous kernel. We employ the asymptotics as an initial approximation in the proposed method of successive approximations to the desired solutions. We prove the existence of a continuous solution depending on free parameters and establish sufficient conditions for the existence of a unique continuous solution. We illustrate the proved existence theorems with examples.