Generalizations to the converse of contraction mapping principle

Citation

Wong, James Sai-Wing (1964) Generalizations to the converse of contraction mapping principle. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-02102004-095235

Abstract

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Let X be a non-empty abstract set and [...] be a commutative semi-group of operators defined on X into itself. [...] is called a contractive semi-group on X if there exists a metric [...] on X such that for each [...] for all x, [...], where [...]. We find sufficient conditions on [...] in order that [...] be contractive on X. In the case when [...] is generated by a finite number of mutually commuting mappings [...] , [...] ,. . . . , [...], 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 [...] , [...] , . . . , [...] defined on X into itself such that each iteration [....] . . . . . . . . [...] ( where [...] , [...] , . . . , [...] are non-negative integers not all equal to zero) possesses a unique fixed point which is common to every choice of [...] , [...] , . . . , [...] . Then for each [...] (0 , 1), there exists a complete metric [...] on X such that [...] for [...], and for all [...]. 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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