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Zbl 0964.34004
Kilbas, A.A.; Bonilla, B.; Trujillo, J.J.
Existence and uniqueness theorems for nonlinear fractional differential equations.
(English)
[J] Demonstr. Math. 33, No.3, 583-602 (2000). ISSN 0420-1213

Summary: The authors study the following Cauchy-type problem for the nonlinear differential equation of fractional order $\alpha\in \bbfC$, $\text{Re}(\alpha)> 0$, $$(D^\alpha_{a+}y)(x)= f[x, y(x)],\quad n-1< \text{Re}(\alpha)\le n,\quad n= -[-\text{Re}(\alpha)],$$ $$(D^{\alpha- k}_{a+} y)(a+)= b_k,\quad b_k\in \bbfC,\quad k= 1,2,\dots, n,$$ containing the Riemann-Liouville fractional derivative $D^\alpha_{a+}y$, on a finite interval $[a,b]$ of the real axis $\bbfR= (-\infty, \infty)$ in the space of summable functions $L(a,b)$. An equivalence of this problem and a nonlinear Volterra integral equation are established. The existence and uniqueness of the solution $y(x)$ to the above Cauchy-type problem are proved by using the method of successive approximations. Corresponding assertions for the ordinary differential equations are presented. Examples are given.
MSC 2000:
*34A25 Analytical theory of ODE
26A33 Fractional derivatives and integrals (real functions)
34A12 Initial value problems for ODE

Keywords: nonlinear differential equation; fractional order; Riemann-Liouville fractional derivative; nonlinear Volterra integral equation; existence; uniqueness; solution; successive approximations

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