The problem of determining the one-dimensional kernel of the viscoelasticity equation

We consider the integrodifferential system of viscoelasticity equations. The direct problem consists in determining the displacement vector from the initial boundary value problem for this system. Under the assumption that the coefficients of the equation depend only on one space variable $x_3$, the system is reduced to an equation for one component $u_1(x_3,t)$. For this equation, we investigate the problem of finding the kernel belonging to the integral part of the equation. For its determination, an additional condition is given on $u_1(x_3,t)$ for $x_3=0$. The inverse problem is replaced by an equivalent system of integral equations for unknown functions. To this system, we apply the contraction mapping principle. A theorem of global unique solvability is proved and a stability estimate of a solution to the inverse problem is obtained.