Generalizations to the Converse of Contraction Mapping Principle

Author: Wong, James Sai Wing

Year: 1965

Degree: Dissertation (Ph.D.)

Advisor: Bohnenblust, Henri Frederic

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/DYFP-7769

Abstract

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.).

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