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On the properties of a Cauchy-type problem for an abstract differential equation with fractional derivatives. (English. Russian original) Zbl 1160.34053
Math. Notes 82, No. 5, 596-607 (2007); translation from Mat. Zametki 82, No. 5, 665-677 (2007).
The author studies Cauchy-type problems for fractional differential equations of Riemann-Liouville type of the form \[ D^\alpha D^\beta u(t) = Au(t). \] His particular interest is in the stability of solutions under perturbations of \(A\), the right hand side of the equation. He considers perturbations which take the form of a bounded operator \(P\). The paper also considers the effect of an inhomogeneous forcing term \(f\) on the right hand side and of a fractional power cosine generator for \(A\). Proofs of well-posedness of the problem and expressions for the solution are given.

MSC:
34G10 Linear differential equations in abstract spaces
34A05 Explicit solutions, first integrals of ordinary differential equations
26A33 Fractional derivatives and integrals
47D09 Operator sine and cosine functions and higher-order Cauchy problems
47N20 Applications of operator theory to differential and integral equations
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
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