Commuting contractive families
Tom 231 / 2015
Fundamenta Mathematicae 231 (2015), 225-272
MSC: Primary 47H09; Secondary 54E50.
DOI: 10.4064/fm231-3-2
Streszczenie
A family $f_1,\ldots ,f_n$ of operators on a complete metric space $X$ is called contractive if there exists a positive $\lambda < 1$ such that for any $x,y$ in $X$ we have $d(f_i(x),f_i(y)) \leq \lambda d(x,y)$ for some $i$. Austin conjectured that any commuting contractive family of operators has a common fixed point, and he proved this for the case of two operators. We show that Austin's conjecture is true for three operators, provided that $\lambda $ is sufficiently small.
