A second-order boundary value problem with nonlinear and mixed two-point boundary conditions is considered,Lx=f(t,x,x′),t∈(a,b),g(x(a),x(b),x′(a),x′(b))=0,x(b)=x(a)in whichLis a formally self-adjoint second-order differential operator. Under appropriate assumptions onL,f, andg, existence and uniqueness of solutions is established by the method of upper and lower solutions and Leray-Schauder degree theory. The general quasilinearization method is then applied to this problem. Two monotone sequences converging quadratically to the unique solution are constructed.