Boundary value problems with compatible boundary conditions

If Y is a subset of the space R-n x R-n, we call a pair of continuous functions U, V Y-compatible, if they map the space R-n into itself and satisfy Ux (.) Vy >= 0, for all (x, y) is an element of Y with x (.) y >= 0. (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential n-dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its domain, Brouwer's fixed point theorem is applied to the composition of the consequent mapping with some projections and a one-parameter family of fixed points P-delta is obtained. Then passing to the limits as delta tends to zero the so-obtained accumulation points are solutions of the problem.