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Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations. (English) Zbl 1214.34007

The authors investigate the existence and uniqueness of solutions for the multi-point boundary value problem for fractional differential equations of the form

${D}_{t}^{\alpha }y\left(t\right)=f\left(t,y\left(t\right),{D}_{t}^{\beta }y\left(t\right)\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}t\in \left(0,1\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$
$y\left(0\right)=0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{D}_{t}^{\beta }y\left(1\right)-\sum _{i=1}^{m-2}{\zeta }_{i}{D}_{t}^{\beta }y\left({\xi }_{i}\right)={y}_{0},\phantom{\rule{2.em}{0ex}}\left(2\right)$

where $1<\alpha \le 2$, $0<\beta <1$, $0<{\xi }_{i}<1,$ $i=1,2,\cdots ,m-2$, ${\xi }_{i}\ge 0$ with $\gamma ={\sum }_{i=1}^{m-2}{\zeta }_{i}{\xi }_{i}^{\alpha -\beta -1}<1$ and ${D}_{t}^{\alpha }$ represents the Riemann-Liouville fractional derivative.

The main tool used by the authors is based on fixed point theory. Specifically, they use the contraction mapping principle and the Schauder fixed point theorem.

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