One case of appearance of positive solutions of delayed discrete equations. (Q851591)

The authors study the existence of a positive solution of the nonlinear delay discrete equation \[ \Delta u(k+n)=f\big (k,u(k),u(k+1),\dots ,u(k+n)\big ), \quad k\in {\mathbb Z},\;k\geq a. \] They derive a sharp sufficient condition for the existence of such solution. The result is a special case with \(b(k)\equiv 0\) and \(c(k):=\sqrt {k}\, \big ( \frac {n}{n+1} \big )^k\) of the existence and boundedness result.