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Fractional differential equations with a Krasnoselskii-Krein type condition. (English) Zbl 1181.34008
Summary: We consider an initial value problem for a fractional differential equation of Caputo type. The convergence of the Picard successive approximations is established by first showing that the Caputo derivatives of these approximations converge.

##### MSC:
 34A08 Fractional differential equations 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions
Full Text:
##### References:
 [1] Agarwal, R. P.; Lakshmikantham, V.: Uniqueness and nonuniqueness criteria for ordinary differential equations. (1993) · Zbl 0785.34003 [2] Kooi, O.: The method of successive approximations and a uniqueness theorem of Krasnoselskii--Krein in the theory of differential equations. Nederi. akad. Wetensch, proc. Ser. A61; indag. Math. 20, 322-327 (1958) [3] Krasnoselskii, M. A.; Krein, S. G.: On a class of uniqueness theorems for the equation y’=$f(x,y)$, usphe. Mat. nauk (N.S). 11, No. 1, 209-213 (1956) [4] Lakshmikantham, V.; Leela, S.; Devi, J. Vasundhara: Theory of fractional dynamical systems. (2009) · Zbl 1188.37002 [5] Lakshmikantham, V.; Leela, S.: Krasnoselskii--Krein type uniqueness result for fractional differential equations. Nonlinear anal. TMA 71, No. 7--8, 3421-3424 (2009) · Zbl 1177.34004