OnM-functions and nonlinear relaxation methods

Globally convergent nonlinear relaxation methods are considered for a class of nonlinear boundary value problems (BVPs), where the discretizations are continuousM-functions.It is shown that the equations with one variable occurring in the nonlinear relaxation methods can always be solved by Newton's method combined with the Bisection method. The nonlinear relaxation methods are used to get an initial approximation in the domain of attraction of Newton's method. Numerical examples are given.