The problem of finding the kernels in the system of integro-differential Maxwell's equations

We pose the direct and inverse problem of finding the electromagnetic field and the diagonal memory matrix for the reduced canonical system of integro-differential Maxwell's equations. The problems are replaced by a closed system of Volterra-type integral equations of the second kind with respect to the Fourier transform in the variables $x_1$ and $x_2$ of the solution to the direct problem and the unknowns of the inverse problem. To this system, we then apply the method of contraction mapping in the space of continuous functions with a weighted norm. Thus, we prove the global existence and uniqueness theorems for solutions to the posed problems.