Inverse problem for the viscoelastic equation with additional information of special form

The one-dimensional inverse problem of determining the kernel of the integral term of the integro-differential viscoelasticity equation with constant density and constant Lame coefficients is considered. Firstly, the direct problem is studied and equivalent integral equation for the desired function $u(x,t)$ together with the necessary conditions for this problem are obtained. Secondly, the inverse problem of determining the kernel of the integral term is studied. Using the additional condition, the inverse problem is replaced by an equivalent system of integral equations for unknown functions. The contraction mapping principle is applied to the system of integral equations in the space of continuous functions with weighted norms. Theorem of global unique solvability of the inverse problem is proved.