Citation
Wong, James Sai Wing (1965) Generalizations to the Converse of Contraction Mapping Principle. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/DYFP7769. https://resolver.caltech.edu/CaltechETD:etd02102004095235
Abstract
Let X be a nonempty abstract set and S be a commutative semigroup of operators defined on X into itself. S is called a contractive semigroup 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 T_{1}, T_{2}..., T_{n}, 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 T_{1}, T_{2}..., T_{n} defined on X into itself such that each iteration T_{1}^{k1}, T_{2}..., T_{n}^{kn} ( where k_{1}, k_{2}, ..., k_{n} are nonnegative integers not all equal to zero) possesses a unique fixed point which is common to every choice of k_{1}, k_{2}, ..., k_{n}. Then for each λε(0,1), there exists a complete metric ρ on X such that ρ(T_{i}x,T_{i}y)≤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), 4143.).
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  (Mathematics) ; 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): 

Group:  Caltech Distinguished Alumni Award 
Thesis Committee: 

Defense Date:  1 May 1964 
Record Number:  CaltechETD:etd02102004095235 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd02102004095235 
DOI:  10.7907/DYFP7769 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  588 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  13 Feb 2004 
Last Modified:  10 Feb 2024 00:08 
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