Abstract: By using the fixed-point index theory of cone mapping, we show the optimal conditions for the existence of positive solutions for second order discrete periodic boundary value problems
$\left\{ {\begin{array}{*{20}{l}} {\Delta^2 y(n-1)+a(n)y(n)=g(n)f(y(n)),}&{n\in[1,N]_{\mathbb{Z}},}\\ {y(0)=y(N), \;\;\;\Delta y(0)=\Delta y(N)}&{} \end{array}} \right.$
with vanishing Green’s function, where $[1,N]_{\mathbb{Z}}=\{1,2,\cdot\cdot\cdot, $$N\},\,f:[1,N]_{\mathbb{Z}}\times\mathbb{R}^+\rightarrow\mathbb{R}^+$ is continuous, $a: [1,N]_{\mathbb{Z}}\rightarrow(0,+\infty),$ and $\mathop {\max }\limits_{n \in {{[1,N]}_{\mathbb{Z}}}} a(n)\leqslant4\sin^2(\frac\pi{2N}),\,g\in C([1,N]_{\mathbb{Z}},\mathbb{R}^+), $$\mathbb{R}^+:=[0,\infty)$.

Key words: periodic boundary value problem, positive solution, nonnegative Green’s function, fixed-point index