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Correctness of Cauchy-type problems for abstract differential equations with fractional derivatives. (English. Russian original) Zbl 1177.47090
Russ. Math. 53, No. 9, 10-19 (2009); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 9, 13-24 (2009).
The author studies the well-posedness of several Cauchy-type problems for fractional differential equations in a Banach space \(X\) of functions defined on \(\mathbb{R}^1_+\). One of these problems is for the equation \[ D^\alpha m_k D^\beta u(t)=t^\gamma Au(t), \;t>0, \] with the initial conditions \[ \lim_{t\to 0}D^{\beta-1}u(t)=u_0, \;\lim_{t\to 0}D^{\alpha-1}(t^k D^\beta u(t))=u_1, \] where \(m_k\) is the operator of multiplication by \(t^k\), \(D^\alpha\) is the Riemann–Liouville fractional differentiation operator and \(A\) is an operator bounded in \(X\).

MSC:
47N20 Applications of operator theory to differential and integral equations
26A33 Fractional derivatives and integrals
34G10 Linear differential equations in abstract spaces
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
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