Solvability for fractional differential inclusions with fractional nonseparated boundary conditions (Q2792426)

The paper studies the following boundary value problem for a fractional differential inclusion NEWLINENEWLINE\[NEWLINE D^q_Cx(t)\in F(t,x(t)), \quad t\in [0,T], NEWLINE\]NEWLINE NEWLINENEWLINE\[NEWLINE a_1x(0)+b_1x(T)=c_1\int_0^Tg(s,x(s))ds,\quad a_2D^{\gamma }_Cx(0)+b_2D^{\gamma }_Cx(T)=c_2\int_0^Th(s,x(s))ds, NEWLINE\]NEWLINE NEWLINEwhere \(D^q_C\) is the Caputo fractional derivative of order \(q\), \(q\in (1,2]\), \(\gamma \in (0,1)\), \(F:[0,T]\times {\mathbb R}\to \mathcal{P}({\mathbb R})\) is a set-valued map, \(g,h:[0,T]\times \mathbb{R}\to \mathbb{R}\) are given maps and \(a_i,b_i,c_i\in \mathbb{R}\), \(i=1,2\).NEWLINENEWLINEThe authors present three existence results for the problem considered. These results are based on a nonlinear alternative of Leray-Schauder type, on the Bressan-Colombo selection theorem for lower semicontinuous set-valued maps with decomposable values and on the Covitz and Nadler set-valued contraction principle.