Mean value theorem for boundary value problems with given upper and lower solutions

An approach based on successive application of the mean value theorem or, equivalently, a successive linear interpolation that excludes extrapolation, is described for two-point boundary value problem (BVP) associated with nonlinear ordinary differential equations (ODEs). The approach is applied to solve numerically a two-point singular BVP associated with a second-order nonlinear ODE which is a mathematical model in membrane response of a spherical cap that arises in nonlinear mechanics. The upper and lower bounds on solution for the foregoing second-order ODE are assumed known analytically. Other possible methods such as the successive bisection for the BVP associated with second-order nonlinear ODE and a multivariable Taylor series for the second or higher-order nonlinear ODEs are also discussed to solve two-point BVP. The scope/limitation of the later methods and other possible higher-order methods in the present context are stressed.