Fractal Fract. 2021, 5(1), 20; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5010020 - 02 Mar 2021
Abstract
The aim of this work is to find by the methods of the Laplace transform the conditions for the existence of a strongly continuous resolving family of operators for a linear homogeneous equation in a Banach space with the distributed Gerasimov–Caputo fractional derivative
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The aim of this work is to find by the methods of the Laplace transform the conditions for the existence of a strongly continuous resolving family of operators for a linear homogeneous equation in a Banach space with the distributed Gerasimov–Caputo fractional derivative and with a closed densely defined operator A in the right-hand side. It is proved that the existence of a resolving family of operators for such equation implies the belonging of the operator A to the class , which is defined here. It is also shown that from the continuity of a resolving family of operators at the boundedness of A follows. The existence of a resolving family is shown for and for the upper limit of the integration in the distributed derivative not greater than 2. As corollary, we obtain the existence of a unique solution for the Cauchy problem to the equation of such class. These results are used for the investigation of the initial boundary value problems unique solvability for a class of partial differential equations of the distributed order with respect to time.
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(This article belongs to the Special Issue Fractional Order Systems: Deterministic and Stochastic Analysis)