The one-dimensional integro-differential equation arising in the theory of viscoelasticity with constant density and Lam´e coefficients is considered. The direct problem is to determine the displacement function from the initial boundary-value problem for this equation, provided that the initial conditions are zero. The spatial domain is the closed interval [0, l], and the boundary condition is given by the stress function in the form of a concentrated perturbation source at the left endpoint of this interval and as zero at the right endpoint. For the direct problem, we study the inverse problem of determining the kernel appearing in the integral term of the equation. To find it, we introduce an additional condition for the displacement function at x = 0. The inverse problem is replaced by an equivalent system of integral equations for the 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.

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The problem of determining the convolutional kernel $k(t,x)$, $t>0$, $x \in {\Bbb R}$, included in a hyperbolic integro-differential equation of the second order, is investigated in a domain bounded by a variable $z$ and having weakly horizontal heterogeneity. It is assumed that this kernel weakly depends on the variable $x$ and decomposes into a power series by degrees of a small parameter $\varepsilon$. A method for finding the first two coefficients $k_{0}(t)$, $k_{1}(t)$ of this expansion is constructed according to the given first two moments in the variable $x$ of the solution of the direct problem at $z=0$.

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