Approximations and rapid convergence of solutions of nonlinear three point boundary value problems

The method of quasilinearization for second order differential equation-x″=f(t,x,x′),t∈[0,1],with nonlinear nonlocal three-point boundary conditionsx(0)=0,x′(1)=g(x(η)),0<η⩽1,is developed. A monotone sequence of solutions converging uniformly and rapidly to a unique solution of the problem is obtained.