Multiple positive solutions for discrete nonlocal boundary value problems (Q879035)

Consider the second order difference equation \[ u(k+1)-2u(k)+u(k-1)+f(k,u(k))=0,\quad k\in \mathbb{Z}_{1,T}\tag{\(*\)} \] subject to one of the following discrete nonlocal boundary conditions: (i) \(u(0)-\beta \Delta u(0)=0,\) \(u(T+1)=\alpha u(\ell),\) or (ii) \(\Delta u(0)=0,\) \(u(T+1)=\alpha u(\ell).\) Here \(T\in \{4,5,\dots\}\) is fixed, \(0<\alpha <1,\beta >0,\) \(\ell \in \mathbb{Z}_{2,T-1},\) and \(f:\mathbb{Z}_{0,T+1}\times [ 0,\infty )\rightarrow \mathbb{R}\) is continuous, where \(\mathbb{Z}_{a,b}:=[a,b]\cap \mathbb{Z}\) for any \(a,b\in \mathbb{Z}\) with \(a\leq b.\) By using a fixed point theorem, the authors establish sufficient conditions for the existence of multiple positive solutions of the boundary value problems \{(\(*\)), (i)\} and \{(\(*\)), (ii)\}.