On contraction principles for mappings defined on a metric space with a directed graph

Abstract

We discuss Jachymski's fixed point theorems Jachymski (2008) [4] for contractions defined on a metric space with a directed graph. We use the chain approach studied by Turinici to prove the results without verifying the existence of the initial point in the Picard sequence. The technique we use defines an appropriate metric or a quasi-metric which relaxes the conditions attached to the directed graph. Using this technique, we not only deduce Jachymski's results but obtain new supplement results as well. We also discuss Hutchinson's theorem for a finite family of contractions defined on a directed graph which was recently proved by Gwóźdź-Łukawska and Jachymski (2009) [2].