# The Cauchy problem for distributed order equations in Banach spaces

• Streletskaya Elizaveta M., wwugazi@gmail.com Chelyabinsk State University, 129 Kashirin Brothers Street, Chelyabinsk, 454001, Russia
• Fedorov Vladimir E., kar@csu.ru Chelyabinsk State University, 129 Kashirin Brothers Street, Chelyabinsk, Russia 454001;Shadrinsk State Pedagogical University, 3 Karl Liebknecht Street, Shadrinsk, Russia 641870; South Ural State University (National Research University), 76 Lenin Avenue, Chelyabinsk, Russia 454080
• Debbouche Amar, amar_debbouche@yahoo.fr Guelma University, 401, Guelma, Algeria 24000
Keywords: evolution equation, fractional Gerasimov-Caputo derivative, Cauchy problem, distributed order equation

### Abstract

The Cauchy problem for a distributed order equation in a Banach space with the fractional Gerasimov–Caputo derivative and a linear bounded operator in the right-hand side is studied. Existence and uniqueness conditions for the problem solution in the space of exponentially growing functions are found by the methods of the Laplace transformation theory. The solution is presented in the form of a contour integral of the bounded operator resolvent with a complex argument determined by the form of the distributed derivative. The analyticity of the solution in the right half-plane of the complex plane is proved. The general result is applied to the research of the Cauchy problem for an integro-differential system of equations with right-hand side in the form of composition of an integral operator with respect to the spatial variables and the linear transformation of the unknown vector-function.

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Streletskaya, E., Fedorov, V. and Debbouche, A. (&nbsp;) “The Cauchy problem for distributed order equations in Banach spaces”, Mathematical notes of NEFU, 25(1), pp. 63-72. doi: https://doi.org/10.25587/SVFU.2018.1.12769.
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Mathematics