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Existence and uniqueness of solution for a class of nonlinear fractional order differential equations. (English) Zbl 1251.34010
Summary: 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
##### Keywords:
Caputo fractional derivative; contraction principle
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##### References:
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