Existence results for fractional integral inclusions via nonlinear alternative for contractive maps (Q467824)

The authors consider a fractional differential inclusion with nonlocal boundary conditions of integral-type. More specifically, they consider the problem \[ -D^{\alpha}x(t)\in AF(t,x(t))+BI^{\beta}G(t,x(t)),\;2<\alpha\leq 3,\;t\in[0,1] \] subject to the boundary conditions \[ D^{\delta}x(0)=D^{\delta+1}x(0)=0 \] and \[ D^{\delta}x(1)=\int_0^{\eta}D^{\delta}x(s)\;ds\text{, }0<\eta<1. \] In the above it holds that \(0<\delta\leq 1\), \(2<\alpha-\delta<3\), and \(\beta>0\). Moreover, the type of fractional derivative considered in this paper is the well-known Riemann-Liouville type. The authors consider a couple of existence results. In the first of these it is assumed that the maps \(F\) and \(G\) are convex-valued, whereas in the second it is not necessarily assumed that this condition holds; rather, in this latter case a lower semicontinuity-type condition is imposed. Furthermore, these results are proved under some additional auxiliary assumptions on the maps \(F\) and \(G\) appearing above. The paper concludes with a numerical example to explicate the results proved in the paper. All in all, this is a pleasantly written paper that should interest those with dual interests of fractional-order boundary value problems and differential inclusions.