INTRODUCTION IN THIS paper we shall be concerned with the existence and uniqueness of solutions to three point boundary value problems associated with the differential equation y”’ = f(x, Y, Y’S Y”), (1.1) where f(x, y, z, w) is assumed to be continuous on a subset of R4. Initial-value problems associated with (1.1) exist, and are unique in the interval [x1, x,]. A monotonicity restriction on f(x, y, z, w) ensures that the following boundary value problems Y “’ = J-(x, Y, Y’, Y”), y(xJ = y, ; y(xJ = y,; y(‘)(x,) = m (i = 1,2), (12i) Y”’ = J-(x, Y, Y', Y), y(x)=y * y”)(x)=m2 2’ 2 9 Y(x,) = Y, (i = 421, (le3i)