Improved convergence theorems of Newton's method designed for the numerical verification for solutions of differential equations

Though the convergence theorem of simplified Newton's method is an excellent general principle for the numerical verification of isolated solutions of differential equations, it is not always good from the viewpoint of computational efficiency, in particular when we use finite element solutions as approximate solutions. We improve the theorem to overcome this point. Some numerical examples on the nonlinear elliptic equations show that the remarkable increase of computational efficiency is achieved by our improvement.