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**Cauchy type problems for fractional differential equations.**
*(English)*
Zbl 1490.34067

Summary: While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann-Liouville derivatives is less understood. In this paper, we propose new type initial, inner, and inner-boundary value problems for fractional differential equations with the Riemann-Liouville derivatives. The results on the existence and uniqueness are proved, and conditions on the solvability are found. The well-posedness of the new type of initial, inner, and inner-boundary conditions is also discussed. Moreover, we give explicit formulas for the solutions. As an application fractional partial differential equations for general positive operators are studied.

### MSC:

34G20 | Nonlinear differential equations in abstract spaces |

34A08 | Fractional ordinary differential equations |

34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |

35R11 | Fractional partial differential equations |

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\textit{E. Karimov} et al., Integral Transforms Spec. Funct. 33, No. 1, 47--64 (2022; Zbl 1490.34067)

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