The authors prove existence, uniqueness and continuous dependence on the initial data for the problem
in the space
where is a real Banach space, is a continuous function and is the fractional derivative of of order (in the sense of Riemann-Liouville). They also discuss flow invariance and inequalities in cones.
Note added by the reviewer: For previous results on existence (and also asymptotic behavior of solutions) for a similar problem, we refer the reader to the papers by the present reviewer with K. M. Furati: J. Fractional Calc. 26, 43–51 (2004; Zbl 1101.34001); J. Fractional Calc. 28, 23–42 (2005; Zbl 1131.26304); Nonlinear Anal., Theory Methods Appl. 62, No. 6 (A), 1025–1036 (2005; Zbl 1078.34028), J. Math. Anal. Appl. 332, No. 1, 441–454 (2007; Zbl 1121.34055).