The one-dimensional inverse problem for the equation of viscoelasticity in a bounded domain.

One-dimensional integro-differential equation, which arises in the theory of viscoelasticity with density $\rho=\rho(x)$ and Lame coefficients $\mu=\mu(x), \ \lambda=\lambda(x)$ is considered. This problem is studied in a bounded domain with respect to $x$, exactly on segment $[0, l]$. The initial conditions are zero. The boundary conditions are a function of the stress at the left end of segment $[0, l]$ in the form of a concentrated source of perturbation, and on the right - zero. For the direct problem we study the inverse problem of determining the kernel belonging to the integral term of the equation, for supplementary information about the function of the displacement at $x = 0$. The inverse problem is replaced by an equivalent system of integral equations for the unknown functions. To the system in the space of continuous functions with weighted norms, the principle of contraction mappings is applied. Theorems global unique solvability and stability of the solution of the inverse problem are proved.