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 ] subject to the separated boundary conditions p 0 x ( 0 ) − q 0 x ′ ( 0 ) = a , p 1 x ( 1 ) + q 1 x ′ ( 1 ) = b , is developed. A monotone sequence of solutions of linear problems converging uniformly and quadratically to a solution of the problem is obtained.