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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 \[ \begin{cases} D^{q}x=f(t,x), \\ x(t)(t-t_{0})^{1-q}| _{t=t_{0}}=x^{0},\;0<q<1 \end{cases} \] 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).

MSC:

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
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