A fractional finite difference inclusion (Q2792431)

The authors consider the fractional difference inclusion \(\Delta_{\mu-2}^\mu x(t)\in F(t,x(t),\Delta x(t))\), \(t\in\{\mu-2,\mu-1,\dots,b+\mu+2\}\), subject to the boundary conditions \(\Delta x(b+\mu)=A\), \(x(\mu-2)=B\), where \(1<\mu\leq 2\), \(A,B\in\mathbb{R}\) and \(F\) is a compact-valued multifunction. Sufficient conditions are established which guarantee the existence of a solution. The key role in the proof is played by the following fixed point theorem: Each closed-valued contraction multifunction on a complete metric space has a fixed point.