Solvability of a nonlinear Neumann problem for systems arising from a burglary model

We study a one dimensional version of a problem that arises in mathematical modeling for burglary of houses. Our approach uses techniques inspired by coincidence degree, see Mawhin (1979). A priori estimates are obtained through a somewhat unusual combination of estimates based upon maximum or minimum properties and on L1-estimates of the type introduced by Ward, see Ward (1981) in some periodic problems.