Application of implicit function theorems to existence of solutions to ordinary differential equations with nonlocal boundary conditions

We consider the problem y = a(x, y)(y − b), y(0) = 0, y (1) = g y(ξ), y (ξ ) (0 < x < 1, ξ fixed in (0, 1)) as a model of steady-state heat conduction in a rod when the heat flux at the end x = 1 is determined by observation of the temperature and heat flux at some interior point ξ . We establish conditions sufficient for existence, uniqueness, and positivity of solutions. Existence and uniqueness for a semi-infinite problem with a boundary condition of this form are also examined.  2002 Elsevier Science (USA). All rights reserved