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Zbl 1146.34042
Lakshmikantham, V.; Vasundhara Devi, J.
Theory of fractional differential equations in a Banach space.
(English)
[J] Eur. J. Pure Appl. Math. 1, No. 1, 38-45, electronic only (2008). ISSN 1307-5543/e

The authors prove existence, uniqueness and continuous dependence on the initial data for the problem $$\cases D^{q}x=f(t,x), \\ x(t)(t-t_{0})^{1-q}\vert _{t=t_{0}}=x^{0},\;0<q<1 \endcases$$ 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. \par 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 {\it 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).
[Nasser-eddine Tatar (Dhahran)]
MSC 2000:
*34G20 Nonlinear ODE in abstract spaces
34A12 Initial value problems for ODE
34A40 Differential inequalities (ODE)

Keywords: basic existence theory; flow invariance; fractional derivative

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