Existence of positive solutions for difference systems coming from a model for burglary

Authors: TIANLAN CHEN, RUYUN MA

Abstract: In this paper, we use the Brouwer degree to prove existence results of positive solutions for the following difference systems: $$\aligned &{D}_k\Delta^2(A_{k-1}-A^0_{k-1})-(A_{k}-A^0_{k})+N_kf(k, A_{k})=0,\ \ k\in[2, n-1]_\mathbb{Z},\\ &\Delta^2N_{k-1}+\Delta[g(k, A_{k}, \Delta A_{k-1})N_k]-w^2(N_k-1)=0,\ \ k\in[2, n-1]_\mathbb{Z},\\ &\Delta A_{1}=0=\Delta A_{n-1},\ \ \Delta N_{1}=0=\Delta N_{n-1}, \endaligned\eqno $$ where the assumptions on $w,\ D_k, A_k^0, f$, and $g$ are motivated by some mathematical models for the burglary of houses.

Keywords: Neumann problems, Brouwer degree, positive solution, models for house burglary

Full Text: PDF