<p>The work is devoted to abstract integro-differential equations, which are operator models of problems in the theory of viscoelasticity. The functions of the kernels of integral operators can be, in particular, sums of decreasing exponentials or sums of fractional exponential Rabotnov functions with positive coefficients, which are widely used in the theory of viscoelasticity. A method for reducing the original initial problem for a model integro-differential equation with operator coefficients in a Hilbert space to the Cauchy problem for a first-order differential equation is introduced. The exponential decay of solutions under known assumptions for the kernels of integral operators is established. Based on the results obtained, the correct solvability of the original initial value problem for the Volterra integro-differential equation with corresponding estimates for the solution is established.</p>