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## Turkish Journal of Mathematics

#### DOI

10.3906/mat-1805-11

#### Abstract

We show existence results of positive solutions of Neumann problems for a discrete system: \aligned &\eta\Delta^2(A_{k-1}-A^0_{k-1})-A_{k}+A^0_{k}+N_kA_{k}=0,\ \ k\in[2, n-1]_\mathbb{Z},\\ &\Delta\big(\Delta N_{k-1}-2N_k\frac{\Delta A_{k-1}}{A_{k}}\big)-N_kA_{k}+A^1_{k}-A^0_{k}=0,\ \ k\in[2, n-1]_\mathbb{Z},\\ &\Delta A_{1}=0=\Delta A_{n-1},\ \ \Delta N_{1}=0=\Delta N_{n-1}, \endaligned where the assumptions on $\eta,\ A_k^0$, and $A_k^1$ are motivated by some mathematics models for house burglary. Our results are based on the topological degree theory.

#### Keywords

Neumann boundary value problems, nonconstant positive solutions, topological degree theory

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