A Numerical Verification Method for Solutions of Boundary Value Problems with Local Uniqueness by Banach's Fixed-Point Theorem

Authors:
Nobito Yamamoto
Affiliations:
-
Venue:
SIAM Journal on Numerical Analysis
Year:
1998

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Abstract

In this paper, we propose a method to prove the existence and the local uniqueness of solutions to infinite-dimensional fixed-point equations using computers. Choosing a set which possibly includes a solution, we transform it by an approximate linearization of the operator appearing in the equation. Then we calculate the radii of the transformed set in order to check sufficient conditions for Banach's fixed-point theorem. This method is applied to elliptic problems and numerical examples are given.