Shock Layer Behavior for Vector Boundary Value Problems and Kinks in the Lower and Upper Solutions

We provide sufficient conditions for the existence and approximations of shock layer solutions of the weakly coupled boundary value problem ϵy′′i=ƒi(t, y)y′i+gi(t, y) i=1, ..., n,yi(a)=Ai, yi(b)=Bi, where y, ƒ, and g are n-dimensional real-valued functions and yi, ƒi and gi are the respective vector components. ϵ is a small positive parameter. Our approach is based upon the construction of lower and upper "solutions" which have boundary layer behavior at specific points Ti (i=1, ..., n) in the interval a