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.
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This research has been supported by the National Aeronautics and Space Administration under Grant No. NGR-40-002-015.
This research has been supported by the National Science Foundation under Grant No. GK-2788.
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Bosarge, W.E., Falb, P.L. Infinite dimensional multipoint methods and the solution of two point boundary value problems. Numer. Math. 14, 264–286 (1970). https://doi.org/10.1007/BF02163335
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DOI: https://doi.org/10.1007/BF02163335