Ulam stabilities for the Darboux problem for partial fractional differential inclusions (Q488820)

In this paper, the authors consider the following Darboux problem for fractional partial differential inclusions \[ \begin{cases} &^cD^r_{\theta}u(x,y)\in F(x,y,u(x,y)); (x,y)\in J:=[0,a]\times [0,b], \\ &u(x,0)=\phi(x), x\in [0,a],\\ &u(0,y)=\psi(y), y\in [0,b],\\ &\phi(0)=\psi(0), \end{cases}\tag{1} \] where \(a, b>0\), \(\theta=(0,0)\), \(^cD^r_{\theta}\) is the fractional Caputo derivative of order \(r=(r_1, r_2)\in (0,1]\times (0,1]\), \(F: J\times E\to {\mathcal P}(E)\) is a nonconvex valued set-valued function with nonempty values in a separable Banach space \(E\), \(\phi: [0,a]\to E\), \(\psi: [0,b]\to E\) are given absolutely continuous functions and \({\mathcal P}(E)\) is the family of all nonempty subsets of \(E\). Ulam's type stability concepts are investigated via Covitz-Nadler's fixed point theorem and fractional version of Gronwall's inequality.