Representation of Solutions of Integro-Differential Equations with Kernels Depending on the Parameter

Integro-differential equations with unbounded operator coefficients in a separable Hilbert space are studied. These equations are an abstract form of the Gurtin-Pipkin-type equation, which describes finite-speed propagation of heat in media with memory. A representation of strong solutions of these equations is derived in the form of the sums of series in exponents that correspond to the spectral points of operator-functions that are the symbols of these equations.