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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 $$(\cal D^\alpha - \rho t \cal D^\beta)x(t) = f(t, x(t), \cal D^\gamma x(t)),\quad t \in (0, 1)$$ 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 $\cal D^\alpha$ 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:
34A08Fractional differential equations
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
34B15Nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
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References:
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