Electron. J. Differential Equations,
Vol. 2018 (2018), No. 176, pp. 117.
Initialvalue problems for linear distributedorder
differential equations in Banach spaces
Vladimir E. Fedorov, Elizaveta M. Streletskaya
Abstract:
We solve the Cauchy problem for inhomogeneous distributedorder
equations in a Banach space with a linear bounded operator
in the righthand side, with respect to the distributed Caputo derivative.
First we find the solution by using the unique solvability theorem for
the Cauchy problem. Then the results obtained are applied to the
analysis of a distributedorder system of ordinary differential equations.
Then we study an analogous equation, but with degenerate linear operator
at the distributed derivative, which is called a degenerate equation.
The pair of linear operators in the equation is assumed to be relatively
bounded. For the two types of initialvalue problems, we obtain the existence
and uniqueness of a solution, and derive its form.
Abstract results for the degenerate equations are used in the study
of initialboundary value problems with distributed order in time
equations with polynomials of selfadjoint elliptic differential
operator with respect to the spatial derivative.
Submitted August 13, 2018. Published October 30, 2018.
Math Subject Classifications: 35R11, 34G10, 47D99, 34A08.
Key Words: Distributed order differential equation;
fractional Caputo derivative; differential equation in a Banach space;
degenerate evolution equation; Cauchy problem.
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Vladimir E. Fedorov
Mathematical Analysis Department
Chelyabinsk State University
129 Kashirin Brothers St.
Chelyabinsk, 454001 Russia
email: kar@csu.ru


Elizaveta M. Streletskaya
Mathematical Analysis Department
Chelyabinsk State University
129 Kashirin Brothers St.
Chelyabinsk, 454001 Russia
email: wwugazi@gmail.com

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