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Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation. (English) Zbl 1262.34008
Summary: We investigate the fractional differential equation \[ u^{\prime\prime}+A^c D^\alpha u=f(t, u,^c D^\mu u, u^\prime) \] subject to the boundary conditions \[ u^\prime(0)=0, u(T)+au^\prime(T)=0. \] Here, \(\alpha\in (1,2),\mu\in (0,1)\), \(f\) is a Carathéodory function and \(^c D\) is the Caputo fractional derivative. Existence and uniqueness results for the problem are given. The existence results are proved by the nonlinear Leray-Schauder alternative. We discuss the existence of positive and negative solutions to the problem and properties of their derivatives.

34A08 Fractional ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for 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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