Generalizations to the Converse of Contraction Mapping Principle

Citation

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. https://resolver.caltech.edu/CaltechETD:etd-02102004-095235

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..., 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 T₁, T₂..., T_n defined on X into itself such that each iteration T₁^k₁, T₂..., T_n^k_n ( where k₁, k₂, ..., k_n are non-negative integers not all equal to zero) possesses a unique fixed point which is common to every choice of k₁, k₂, ..., k_n. Then for each λε(0,1), there exists a complete metric ρ on X such that ρ(T_ix,T_iy)≤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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