Geodesic
Cauchy-type problem for an abstract differential equation with fractional derivatives
Matematičeskie zametki, Tome 77 (2005) no. 1, pp. 28-41

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The uniform well-posedness of a Cauchy-type problem with two fractional derivatives and bounded operator $A$ is proved. For an unbounded operator $A$ we present a test for the uniform well-posedness of the problem under consideration consistent with the test for the uniform well-posedness of the Cauchy problem for an equation of second order. 
@article{MZM_2005_77_1_a2,
     author = {A. V. Glushak},
     title = {Cauchy-type problem for an abstract differential equation with fractional derivatives},
     journal = {Matemati\v{c}eskie zametki},
     pages = {28--41},
     publisher = {mathdoc},
     volume = {77},
     number = {1},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_77_1_a2/}
}
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A. V. Glushak. Cauchy-type problem for an abstract differential equation with fractional derivatives. Matematičeskie zametki, Tome 77 (2005) no. 1, pp. 28-41. http://geodesic.mathdoc.fr/item/MZM_2005_77_1_a2/