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Distributed order equations in Banach spaces with sectorial operators. (English) Zbl 1494.34132

Kravchenko, Vladislav V. (ed.) et al., Transmutation operators and applications. Cham: Birkhäuser. Trends Math., 509-538 (2020).
Summary: We study the Cauchy problem for a class of solved with respect to the distributed Gerasimov-Caputo derivative inhomogeneous equations in Banach spaces with a linear unbounded operator, generating an analytic in a sector resolving family of operators. The unique solvability theorem for the Cauchy problem is proved, the form of the solution is found. These results are applied to the research of the Cauchy problem and the Showalter-Sidorov problem for linear inhomogeneous equations in Banach spaces with degenerate operator at the distributed order derivative. In the case of the generation by the pair of operators (at unknown function and its distributed order derivative) of an analytic resolving family of the corresponding degenerate homogeneous equation, we obtain theorems of the existence of a unique solution to such problems, and derive the form of the solution. Abstract results for the degenerate equation are used for research of initial-boundary value problems unique solvability for a class of distributed order in time equations with polynomials of self-adjoint elliptic differential operator with respect to the spatial variables.
For the entire collection see [Zbl 1443.34001].

MSC:

34G10 Linear differential equations in abstract spaces
34A08 Fractional ordinary differential equations
47B12 Sectorial operators
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