Generalizations to the Converse of Contraction Mapping Principle

contraction mapping principle

Wong, James Sai-Wing ; Bohnenblust, Henri Frederic

University:California Institute of Technology

Department:Physics, Mathematics and Astronomy

关键词: contraction mapping principle;   

Others  :  https://thesis.library.caltech.edu/588/1/Wong_jsw_1965.pdf

美国|英语

来源: Caltech THESIS

【 摘 要 】

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.).