On the existence of positive solutions of fourth-order difference equations (Q1763228)

Let \(T\geq 1\) fixed, \(f\in C(\mathbb{R}^+,\mathbb{R}^+)\), \(a:\{1,2,\dots, T+1\}\subset\mathbb{Z}\to \mathbb{R}^+\) and \(\lim_{x\to 0,x> 0}\,x^{-1} f(x)\) and \(\lim_{x\to+\infty}\, x^{-1} f(x)\) be limits that exist. The authors determine eigenvalues \(\lambda\) for which there exist positive solutions \(u\) of the fourth-order difference equation \[ \Delta^4 u(t- 2)- \lambda a(t) f(u(t))= 0,\quad t\in \{2,3,\dots, T+ 2\}\subset\mathbb{Z} \] satisfying the boundary conditions \[ u(0)= \Delta^2 u(0)= u(T+ 2)= \Delta^2 u(T)= 0 \] or \[ u(0)= \Delta^2 u(0)= \Delta u(t+ 1)= \Delta^3 u(T- 1)= 0 \] by means of Krasnosel'skij's fixed point theorem.