An inverse problem of determining the kernel in an integro-differential equation of vibrations of a bounded string

We consider an integro-differential equation of hyperbolic type in the domain $D={(x, t) : 0 < x < l, t > 0}$ bounded in the variable $x$. The direct problem is investigated rst. For the direct problem, the inverse problem of determining the kernel of the integral term of the integro-differential equation is studied on the basis of the available additional information about the solution of the direct problem for $x=0$. Differentiating the obtained integral equation for $u(x, t)$ three times with respect to $t$ and using some additional condition, we reduce the solution of the inverse problem to solving a system of integral equations for unknown functions. The contraction mapping principle is applied to this system in the space of continuous functions with weighted norms. A theorem on the global unique solvability is proved. An estimate for the conditional stability of the solution to the inverse problem is also obtained.