Positive solution for BVPs of fourth order difference equations (Q2493170)

The author studies the existence of a positive solution for the following boundary value problem (BVP) of the nonlinear fourth order difference equation \[ \Delta^4 x(t-2)=a(t)f(x(t)),\quad t\in [2,T], \] \[ x(0)=x(T+2)=0,\quad \Delta^2 x(0)=\Delta^2 x(T)=0, \] where \(f: [0,\infty)\to [0,\infty)\) is continuous, \(a: [2,T]\to [0,\infty)\) is not identical zero. By using of the well-known Guo-Krasnosel'skii fixed point theorem in a cone, an existence result of at least one positive solution for the BVP is obtained in the case that \(f\) is superlinear (\(f_0=0, f_\infty=\infty\)) or sublinear (\(f_0=\infty, f_\infty=0\)), where \(f_0=\lim_{u\to 0^+}\frac{f(u)}{u}\) and \(f_\infty=\lim_{u\to \infty}\frac{f(u)}{u}\).