TY - JOUR

T1 - A theorem for numerical verification on local uniqueness of solutions to fixed-point equations

AU - Yamamoto, Nobito

AU - Nakao, Mitsuhiro T.

AU - Watanabe, Yoshitaka

PY - 2011

Y1 - 2011

N2 - We give a theoretical result with respect to numerical verification of existence and local uniqueness of solutions to fixed-point equations which are supposed to have Fréchet differentiable operators. The theorem is based on Banach's fixed-point theorem and gives sufficient conditions in order that a given set of functions includes a unique solution to the fixed-point equation. The conditions are formulated to apply readily to numerical verification methods. We already derived such a theorem in [11], which is suitable to Nakao's methods on numerical verification for PDEs. The present theorem has a more general form and one may apply it to many kinds of differential equations and integral equations which can be transformed into fixed-point equations.

AB - We give a theoretical result with respect to numerical verification of existence and local uniqueness of solutions to fixed-point equations which are supposed to have Fréchet differentiable operators. The theorem is based on Banach's fixed-point theorem and gives sufficient conditions in order that a given set of functions includes a unique solution to the fixed-point equation. The conditions are formulated to apply readily to numerical verification methods. We already derived such a theorem in [11], which is suitable to Nakao's methods on numerical verification for PDEs. The present theorem has a more general form and one may apply it to many kinds of differential equations and integral equations which can be transformed into fixed-point equations.

KW - Computer-assisted proof

KW - Fixed-point equation

KW - Local uniqueness

KW - Numerical verification

KW - Self-validated computing

UR - http://www.scopus.com/inward/record.url?scp=84855791012&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84855791012&partnerID=8YFLogxK

U2 - 10.1080/01630563.2011.594348

DO - 10.1080/01630563.2011.594348

M3 - Article

AN - SCOPUS:84855791012

SN - 0163-0563

VL - 32

SP - 1190

EP - 1204

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

IS - 11

ER -