Inverse problem for viscoelastic system in a vertically layered medium

For the reduced canonical system of integro-differential equations of
viscoelasticity, direct and inverse problems of determining the velocity
field of elastic waves and the relaxation matrix are posed. The problems
are replaced by a closed system of integral equations of the second kind
of Volterra type with respect to the Fourier transform in the variables
$x_1$, $x_2$ for solving the direct problem and unknowns of the
inverse problem. Further, the method of contraction mappings in the
space of continuous functions with a weighted norm is applied to this
system. Thus, we prove global existence and uniqueness theorems for
solutions to the problems posed.