Wang, Jingjing; Lu, Yanqiong Optimal conditions for the existence of positive solutions to periodic boundary value problems with second order difference equations. (Chinese. English summary) Zbl 1474.39040 J. East China Norm. Univ., Nat. Sci. Ed. 2020, No. 2, 41-49 (2020). Summary: 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 \[\begin{cases}{\Delta^2}y (n-1) + a (n)y (n) = g (n)f(y (n)),\; n \in [1,N]_{\mathbb{Z}},\\ y (0) = y (N),\; \Delta y (0) = \Delta y (N),\end{cases}\] with vanishing Green’s function, where \([1,N]_{\mathbb{Z}} = \{1, 2, \cdots, N\}\), \(f:[1,N]_{\mathbb{Z}} \times {\mathbb{R}^+} \to {\mathbb{R}^+}\) is continuous, \(a:[1,N]_{\mathbb{Z}} \to (0, +\infty)\) and \(\max\limits_{n \in [1,N]_{\mathbb{Z}}} a (n) \le 4{\sin^2} (\frac{\pi}{2N})\), \(g \in C ([1,N]_{\mathbb{Z}}, {\mathbb{R}^+})\), \({\mathbb{R}^+}: = [0,\infty)\). MSC: 39A27 Boundary value problems for difference equations 39A12 Discrete version of topics in analysis 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 39A23 Periodic solutions of difference equations Keywords:periodic boundary value problem; positive solution; nonnegative Green’s function; fixed-point index PDFBibTeX XMLCite \textit{J. Wang} and \textit{Y. Lu}, J. East China Norm. Univ., Nat. Sci. Ed. 2020, No. 2, 41--49 (2020; Zbl 1474.39040) Full Text: DOI