Extremal solutions and relaxation problems for fractional differential inclusions (Q2015328)

Summary: We are concerned with the existence of solutions, Filippov's theorem, and the relaxation theorem of abstract fractional differential inclusions. We consider the following problem: \[ \begin{gathered} ^cD^\alpha y(t)\in F(t,y(t)), \text{ a.e. } t\in J:=[0,b], \qquad y(0)=y_0,\quad y'(0)= y_1,\\ ^cD^\alpha y(t)\in\text{ext}\, F(t,y(t)),\text{ a.e. } t\in J:=[0,b], \qquad y(0)=y_0, \quad y'(0)=y_1, \end{gathered} \] where \(^cD^\alpha\) is the Caputo fractional derivatives, \(\alpha\in(1,2]\), \(F:J\times\mathbb{R}^N\to{\mathcal P}(\mathbb{R}^N)\) is a multifunction, and \(\text{ext}\,F(t,y)\) represents the set of extreme points of \(F(t,y)\). \(({\mathcal P}(\mathbb{R}^N)\) is the family of all nonempty subsets of \(\mathbb{R}^N\).)