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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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##### References:
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