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Existence of solutions for fractional differential inclusions with nonlocal Riemann-Liouville integral boundary conditions. (English) Zbl 1340.34056
The authors consider the existence of solutions for fractional differential inclusions of the form \[ ^cD^qx(t)\in F(t,x(t)),\; 1<q\leq 2, \,t\in [0,1] \] satisfying the following nonlocal Riemann-Liouville integral boundary value conditions \[ x(0)=aI^\beta x(\eta)=a\int_0^\eta \frac {(\eta - s)^{\beta -1}}{\Gamma (\beta)}x(s)ds, \;0<\beta \leq 1, \]
\[ x(1)=bI^\alpha x(\sigma)=b\int_0^\sigma \frac {(\sigma - s)^{\alpha -1}}{\Gamma (\alpha)}x(s)ds, \;0<\alpha \leq 1. \] Here \(^cD^q\) denotes the Caputo fractional derivative of order \(q,\) \(F:[0,1]\times \mathbb {R}\multimap \mathbb {R}\) is a multimap and \(a, b, \eta, \sigma \) are real constants with \(\eta > 0, \, \sigma < 1.\)
Applying some topological methods, the authors consider the following cases: (i) \(F\) is compact convex valued and satisfies Carathéodory type conditions; (ii) \(F\) is compact-valued, measurable and lower semicontinuous in the second variable; (iii) \(F\) is measurable in the first argument and satisfies a Lipschitz-type condition in the second variable.
34A60 Ordinary differential inclusions
34A08 Fractional ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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