On the Convergence of the Initial-Value Adjusting Method for Nonlinear Boundary Value Problems
Taketomo Mitsui · Publications of the Research Institute for Mathematical Sciences · 1980
Here x and X(t, x) are n -dimensional vectors, x(t) is considered as a function J = [a, fc]-»J? and / is an rc-dimensional vector-valued functional on some subset of Q7). Recently Ojika and Kasue [5] have proposed a numerical procedure called initialvalue to solve the multi-point boundary value problem, which is considered to be a powerful algorithm for rather complex constraining conditions. And Ojika [4] has given a proof for convergence of their method. The present paper is devoted to an analysis of the method by the different way from his and to obtaining sufficient conditions for convergence of the iteration. Roughly speaking, for the initial-value method, which can be regarded as a systematical version of the shooting methods, the convergence holds when X and / are sufficiently smooth and the starting value of iteration is taken sufficiently close to the isolated exact one. Moreover, it will be shown that the inverse of the adjusting matrix is a good example of the contractor that has been introduced by Altman [1].