On the Boundary-Layer Solutions for a Class of Vector Problems Containing a Small-Parameter

AbstractDifferential inequality techniques are used to obtain the existence and approximations of boundary layer solutions for the nonlinear vector boundary value problem ϵy″ = ƒ(t, y, y′) for a ≤t≤by(a) = Aandy(b) = B, where ϵ is a small positive parameter. The qualitative behavior of solutions for this class of problems is very sensitive to the growth rates of ƒ with respect to the derivative components y′i, i = 1, … n. A well-known componentwise Nagumo condition is weakened to allow certain components of ƒ, say ƒi, to grow at arbitrarily large rates in certain y′j, j ≠ i. An interesting consequence is the existence of solulions with boundary layers that are transcendentally small in thickness. These results are new and are obtained under relatively mild stability conditions imposed on the data as compared with other studies of this problem. Problems of this type arise in fluid dynamics