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 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 CW(K,a), which is defined here. It is also shown that from the continuity of a resolving family of operators at t=0 the boundedness of A follows. The existence of a resolving family is shown for A∈CW(K,a) 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.
Highlights
In the last couple of decades, a new branch of the theory of differential equations has emerged as equations with distributed fractional derivatives
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
These results were applied to study of initial boundary value problems for some partial differential equations of a distributed order with respect to time
Summary
In the last couple of decades, a new branch of the theory of differential equations has emerged as equations with distributed fractional derivatives. An extension of the theorem on generators of analytic semigroups of operators to the case of distributed order equations is obtained This allows investigating a unique solvability of problem (1) and (2). A theorem on perturbations of generators for analytic resolving family of operators for distributed order Equation (2) is proved. These results were applied to study of initial boundary value problems for some partial differential equations of a distributed order with respect to time. The abstract results are applied to the study of initial boundary value problems for a class of partial differential equations of the distributed order with respect to time
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