A THEOREM FOR NUMERICAL VERIFICATION ON LOCAL UNIQUENESS OF SOLUTIONS TO FIXED-POINT EQUATIONS

作者：Yamamoto Nobito^*; Nakao Mitsuhiro T; Watanabe Yoshitaka

来源：Numerical Functional Analysis and Optimization, 2011, 32(11): 1190-1204.

DOI：10.1080/01630563.2011.594348

• Computer-assisted proof
• Fixed-point equation
• Local uniqueness
• Numerical verification
• Self-validated computing

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 Frechet 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.

• 出版日期2011