On Volterra Integro-Differential Equations with KernelsRepresentable by Stieltjes Integrals

We consider abstract linear inhomogeneous second-order integro-differential equations in a Hilbert space that are defined on the positive half-line and have unbounded coefficients and integral terms of the Volterra convolution type with kernels represented by the Stieltjes integral of a decaying exponential. The equations under study represent an abstract form of partial integro-differential equations arising in the theory of viscoelasticity and have a number of other important applications. Sufficient conditions are found under which the initial problem for the equations under consideration is well-posed solvable in weighted Sobolev spaces, and the localization and structure of the spectrum of operator functions that are symbols of these equations are established. The proposed approach can be used to study other integro-differential equations of this kind.

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