Theory of fractional differential equations in a Banach space. (English) Zbl 1146.34042

The authors prove existence, uniqueness and continuous dependence on the initial data for the problem \[ \begin{cases} D^{q}x=f(t,x), \\ x(t)(t-t_{0})^{1-q}| _{t=t_{0}}=x^{0},\;0<q<1 \end{cases} \] in the space \[ C_{p}([t_{0},t_{0}+a],E) :=\left\{ u:u\in C((t_{0},t_{0}+a],E)\text{ and }(t-t_{0})^{1-q}u(t)\in C([t_{0},t_{0}+a],E)\right\} \] where \(E\) is a real Banach space, \(f\) is a continuous function and \(D^{q}x\) is the fractional derivative of \(x\) of order \(q\) (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).


34G20 Nonlinear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A40 Differential inequalities involving functions of a single real variable