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Solvability of a three-point fractional nonlinear boundary value problem. (English) Zbl 1256.34003

Summary: We study the fractional boundary value problem

${}^{c}{D}_{{0}^{+}}^{q}u\left(t\right)=f\left(t,u\left(t\right)\right),\phantom{\rule{1.em}{0ex}}0
$u\left(0\right)=\alpha {u}^{\text{'}}\left(0\right),\phantom{\rule{1.em}{0ex}}u\left(1\right)=\beta {u}^{\text{'}}\left(\eta \right),$

where $1, $\alpha ,\beta \in ℝ$ and ${}^{c}{D}_{{0}^{+}}^{q}$ denotes Caputo’s fractional derivative. Using Banach contraction principle and Leray-Schauder nonlinear alternative, we prove the existence and uniqueness of solutions. Some examples are given to illustrate our results.

##### MSC:
 34A08 Fractional differential equations 34B10 Nonlocal and multipoint boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations
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