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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

$\left\{\begin{array}{c}{D}^{q}x=f\left(t,x\right),\hfill \\ x\left(t\right){\left(t-{t}_{0}\right)}^{1-q}{|}_{t={t}_{0}}={x}^{0},\phantom{\rule{0.277778em}{0ex}}0

in the space

${C}_{p}\left(\left[{t}_{0},{t}_{0}+a\right],E\right):=\left\{u:u\in C\left(\left({t}_{0},{t}_{0}+a\right],E\right)\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{\left(t-{t}_{0}\right)}^{1-q}u\left(t\right)\in C\left(\left[{t}_{0},{t}_{0}+a\right],E\right)\right\}$

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).

##### MSC:
 34G20 Nonlinear ODE in abstract spaces 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions 34A40 Differential inequalities (ODE)