Journal of East China Normal University(Natural Science) ›› 2020, Vol. 2020 ›› Issue (2): 41-49.doi: 10.3969/j.issn.1000-5641.201811039

• Mathematics • Previous Articles     Next Articles

Optimal conditions for the existence of positive solutions to periodic boundary value problems with second order difference equations

WANG Jingjing, LU Yanqiong   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
  • Received:2018-09-30 Published:2020-03-16

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

CLC Number: