Existence of positive solutions for summation boundary value problem for a fourth-order difference equations (Q2792444)

The author considers the difference summation boundary value problem NEWLINE\[NEWLINE\begin{aligned} & \Delta^4u(t-2)+ a(t)f(u)=0,\quad t\in\{2,3,\dots,T\},\\ & u(0)=\Delta u(0)=\Delta^2(0) =0,\quad u(T+2)=\alpha\sum_1^\eta u(s),\end{aligned}NEWLINE\]NEWLINE where \(f(\cdot)\) is continuous, \(T\geq 5\), \(\eta\in\{4,5,\dots,T-1\}\).NEWLINENEWLINESupposing \(f\) is either sublinear or superlinear, the existence of at least one positive solution is shown. The basic proof instrument is the Guo-Krasnosel'skii fixed point theorem on cones.