We study linear differential equations with the Hilfer derivative and a closed operator A in a Banach space with Cauchy type conditions. We obtain a criterion for the existence of exponentially bounded analytic resolving families of operators in terms of the operator A and its resolvent. For operators satisfying the criterion conditions we establish the unique solvability of Cauchy type problems for linear inhomogeneous equations with the Hilfer derivative and prove a perturbation theorem. General results are applied to establish the unique solvability of an initial-boundary value problem for linearized Boussinesq equations with the Hilfer fractional time-derivative.
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International Mathematical Schools. Vol. 6. Mathematical Schools in Uzbekistan
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Fedorov, V., Apakov, Y. & Skorynin, A. Analytic Resolving Families of Operators for Linear Equations with Hilfer Derivative. J Math Sci 277, 385–402 (2023). https://doi.org/10.1007/s10958-023-06843-x
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DOI: https://doi.org/10.1007/s10958-023-06843-x