Existence and uniqueness of solution for a class of nonlinear fractional order differential equations.

*(English)*Zbl 1251.34010Summary: We discuss the existence and uniqueness of solutions to nonlinear fractional order ordinary differential equations
\[
(\mathcal D^\alpha - \rho t \mathcal D^\beta)x(t) = f(t, x(t), \mathcal D^\gamma x(t)),\quad t \in (0, 1)
\]
with boundary conditions
\[
x(0) = x_0,\quad x(1) = x_1
\]
or satisfying the initial conditions
\[
x(0) = 0,\quad x'(0) = 1,
\]
where \(\mathcal D^\alpha\) denotes the Caputo fractional derivative, \(\rho\) is constant, \(1 < \alpha < 2\), and \(0 < \beta + \gamma \leq \alpha\). Schauder’s fixed-point theorem is used to establish the existence of a solution. Banach’s contraction principle is used to show the uniqueness of the solution under certain conditions on \(f\).

##### MSC:

34A08 | Fractional ordinary differential equations |

34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

47N20 | Applications of operator theory to differential and integral equations |

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\textit{A. Babakhani} and \textit{D. Baleanu}, Abstr. Appl. Anal. 2012, Article ID 632681, 14 p. (2012; Zbl 1251.34010)

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