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Generalizations to the Converse of Contraction Mapping Principle


Wong, James Sai-Wing (1965) Generalizations to the Converse of Contraction Mapping Principle. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/DYFP-7769.


Let X be a non-empty abstract set and S be a commutative semi-group of operators defined on X into itself. S is called a contractive semi-group on X if there exists a metric ρ on X such that for each TεS, T≠I, ρ(Tx,Ty)≤λ(T)ρ(x,y) for all x, yεX, where 0≤λ(T)<1. We find sufficient conditions on S in order that S be contractive on X. In the case when S is generated by a finite number of mutually commuting mappings T1, T2..., Tn, possessing a common unique fixed point in X, these conditions are automatically satisfied. The resulting statement is the following generalization of the converse of contraction mapping principle: Theorem C . Let X be an abstract set with n mutually commuting mappings T1, T2..., Tn defined on X into itself such that each iteration T1k1, T2..., Tnkn ( where k1, k2, ..., kn are non-negative integers not all equal to zero) possesses a unique fixed point which is common to every choice of k1, k2, ..., kn. Then for each λε(0,1), there exists a complete metric ρ on X such that ρ(Tix,Tiy)≤λρ(x,y) for 1≤i≤n, and for all x,yεX. This result reduces to that of C. Bessaga by taking n = 1. ( Rf: C. Bessaga, Colloquim Mathematicum VII (1959), 41-43.).

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:contraction mapping principle
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Caltech Distinguished Alumni Award, 2014
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Bohnenblust, Henri Frederic
Group:Caltech Distinguished Alumni Award
Thesis Committee:
  • Unknown, Unknown
Defense Date:1 May 1964
Record Number:CaltechETD:etd-02102004-095235
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:588
Deposited By: Imported from ETD-db
Deposited On:13 Feb 2004
Last Modified:22 Jul 2021 17:39

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PDF (Wong_jsw_1965.pdf) - Final Version
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