On an integro-differential inclusion of fractional order (Q354235)

The paper deals with the existence of solutions to the following fractional hyperbolic integro-differential inclusion: \[ D^r_cu(x,\,y) \in F(x,\,y,\,u(x,\,y),\,(I^r_0u)(x,\,y))\;\;\; a.e.\; (x,\,y)\in [0,\,T_1]\times [0,\,T_2]; \] \[ u(x,\,0)=\phi(x),\; u(0,\,y)=\psi(y),\;\; (x,\,y)\in [0,\,T_1]\times [0,\,T_2], \] where \(F\) is a set-valued map, \(I^r_0\) is the left sided mixed Riemann-Liouville integral of order \(r=(r_1,\,r_2)\), \(D^r_c\) is the Caputo fractional derivative of order \(r\). The author proves the existence of a solution using a Filippov type approach and, moreover, shows the continuous dependence of the solution on a selection parameter.