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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 $$ ^{c}D_{0^{+}}^{q}u \left( t \right) =f(t, u(t)),\quad 0 < t <1$$ $$u\left( 0 \right) = \alpha u^{\prime} \left( 0 \right) ,\quad u \left( 1\right) =\beta u^{\prime } \left( \eta \right),$$ where $1 < q < 2$, $\alpha, \beta \in \mathbb{R}$ 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.

34A08Fractional differential equations
34B10Nonlocal and multipoint boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
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