Infinite dimensional multipoint methods and the solution of two point boundary value problems

Abstract

Consider the problem of determining the roots of an equation of the formF(χ) =0 whereF maps the Banach spaceX into itself. Convergence theorems for the iterative solution ofF(χ) =0 are proved for multipoint algorithms of the formχ _n+1=χ _n -φ _α (χ _n ),α ≧ 1, where$\phi _\alpha (x) = \sum\limits_{j = 1}^\alpha {(F'_x )^{ - 1} F(x - \phi _{j - 1} (x))}$ and φ₀(χ)=0. The theorems are applied to the solution of two point boundary value problems of the form$\dot y$=f (y, t), g(y(0))+h(y(1))=c. A set {A(t),B,C} of matrices is called boundary compatible if the linear two point boundary value problem$\dot y$=A(t)) y+k (t),B y (0) + C y (1) = d has a unique solution for allk (t) andd. Then, under certain conditions, there are boundary compatible sets such that the problem$\dot y$=f (y, t),g (y (0) ) +h (y (1)) =c has the equivalent integral representation whereΛ andΓ are Green's matrices for the linear problem$\dot y$=A(t)y +k(t),B y (0) +C y (1) =d. Eq. (i) is viewed as an operator equation of the formF (x) =(I-T) (x) = 0 and convergence conditions for the iterative solution of (i) are deduced from the general theorems. Explicit interpretations of the convergence results are given in terms off, g, h and some illustrative numerical examples are presented.