Existence of solutions for a 1-D boundary value problem coming from a model for burglary

Using Leray–Schauder degree arguments, existence results for positive solutions are proved for ordinary differential systems of the form η(x)(A−A0(x))″−A+A0(x)+Nf(x,A)=0,η(x)(A−A0(x))″−A+A0(x)+Nf(x,A)=0,N″+[g(x,A,A′)N]′−ω2(N−1)=0,N″+[g(x,A,A′)N]′−ω2(N−1)=0, with either Neumann or periodic boundary conditions. The form of the system and the assumptions upon ηη, A0A0, ff and gg are motivated by some mathematical models for the burglary of houses. The requested a priori estimates are obtained by some unusual combination of pointwise and L1L1-estimates.