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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

D q x=f(t,x),x(t)(t-t 0 ) 1-q | t=t 0 =x 0 ,0<q<1

in the space

C p ([t 0 ,t 0 +a],E):=u:uC((t 0 ,t 0 +a],E)and(t-t 0 ) 1-q u(t)C([t 0 ,t 0 +a],E)

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).

34G20Nonlinear ODE in abstract spaces
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
34A40Differential inequalities (ODE)