Geodesic
Correct solvability and exponential stability for solutions of Volterra integro-differential equations
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 500 (2021), pp. 62-66

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Abstract integro-differential equations that are operator models of viscoelasticity problems are studied. The kernels of the integral operators can be specified as sums of decreasing exponentials or sums of Rabotnov functions with positive coefficients, which are widely used in viscoelasticity theory. A method is described whereby the original initial value problem for a model integro-differential equation with operator coefficients in a Hilbert space is reduced to the Cauchy problem for a first-order differential equation. Exponential stability of solutions is established under known assumptions on the kernels of the integral operators. The results are used to establish the correct solvability of the original initial value problem for a Volterra integro-differential equation with corresponding solution estimates.
Keywords: Volterra integro-differential equations, linear differential equations in Hilbert spaces, exponential stability. 
@article{DANMA_2021_500_a11,
     author = {N. A. Rautian},
     title = {Correct solvability and exponential stability for solutions of {Volterra} integro-differential equations},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {62--66},
     publisher = {mathdoc},
     volume = {500},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2021_500_a11/}
}
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N. A. Rautian. Correct solvability and exponential stability for solutions of Volterra integro-differential equations. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 500 (2021), pp. 62-66. http://geodesic.mathdoc.fr/item/DANMA_2021_500_a11/