Journal of East China Normal University(Natural Science) >

2020 , Vol. 2020 >Issue 2: 41 - 49

DOI: https://doi.org/10.3969/j.issn.1000-5641.201811039

Mathematics

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

  • WANG Jingjing ,
  • LU Yanqiong
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  • College of Mathematics and Statistics, Northwest Normal University, Lanzhou　730070, China

Received date: 2018-09-30

  Online published: 2020-03-16

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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

Cite this article

WANG Jingjing , LU Yanqiong . Optimal conditions for the existence of positive solutions to periodic boundary value problems with second order difference equations[J]. Journal of East China Normal University(Natural Science), 2020 , 2020(2) : 41 -49 . DOI: 10.3969/j.issn.1000-5641.201811039

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