Positive solutions of nonlinear functional difference equations (Q1600409)

For integer \(t\in[1,T+1]\) the functional difference equation \[ \Delta^2 x(t-1)+ \lambda a(t)f\biggl(x(t), x\bigl(\varphi(t) \bigr) \biggr) =0 \] is considered under both the initial \(x(j)=v(j)\), \(j\in [-r,-1]\), and the boundary conditions \(x(0)=x (T+2)=0\). Under different conditions the existence of a positive solution is proved for small respectively for all positive values \(\lambda\).