Existence results for q-fractional differential inclusions with non-convex right hand side

Abstract

In this paper, we investigate the solutions set for a $q$-fractional differential inclusion (1.1)  $^{c}D_{q}^{\alpha }x(t) \in F ( t,x(t) , ^{c}D_{q}^{\alpha }x(t)),  t\in [0,T]$, with the initial condition (1.2)  $x(0) =x_{0}$ where $q \in (0,1) $ and $\alpha \in ( 0,1]$, $T>0$, $F: [0,T]\times \mathbb{R}\times \mathbb{R} \rightarrow \mathcal{P} (\mathbb{R}) $ is a multi-valued map. Our result is based on the fixed point theorem for multi-valued maps due to Covitz and Nadler. We also establish some Filippov's-type results for the problem (1.1)--(1.2). Finally, an example is presented to illustrate our main results.