A Theorem on Contraction Mapping.
Expert InsightsPublished 1969
A proof that the conclusion of a well-known theorem of Banach holds more generally from a condition of weakly uniformly strict contraction. The following fixpoint theorem is demonstrated: Let (X,d) be a complete metric space and f a mapping of X into itself. If, for all e greater than zero, there exists e' greater than zero such that d(x,y) is between e and e + e' and d(f(x),f(y)) is less than e, then f has a unique fixpoint z. Moreover, for any x in X, the sequence of iterated transforms of x approaches z. Such fixpoint theorems are used in functional analysis and to prove that a differential equation has a unique solution. 6 pp. Ref. (MW)
Document Details
- Copyright: RAND Corporation
- Availability: Web Only
- Year: 1969
- Pages: 6
- DOI: https://doi.org/10.7249/pubs
- Document Number: P-3993
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