The generalized quasilinearization technique for a second order differential equation with separated boundary conditions

The method of upper and lower solutions and the method of quasilinearization for a second order nonlinear differential equation −x″(t)=f(t,x,x′),t∈[0,1]−x″(t)=f(t,x,x′),t∈[0,1] subject to the separated boundary conditions p0x(0)−q0x′(0)=a,p1x(1)+q1x′(1)=bp0x(0)−q0x′(0)=a,p1x(1)+q1x′(1)=b, is developed. A monotone sequence of solutions of linear problems converging uniformly and quadratically to a solution of the problem is obtained.