A sufficient condition for a finite family of continuous functions to be transformed into \(\psi\)-contractions (Q2790807)

As the main result, the authors prove the following theorem for twelve different situations:NEWLINENEWLINE``Let us consider \((X,d)\) a metric space, the continuous functions \( f_{1},\dots,f_{N}:X\rightarrow X\) and a function \(\pi :\Lambda \rightarrow X\), where \(\Lambda =\Lambda \left( \{1,2,\dots,N\}\right) \), such that the condition \(C\) (for the metric \(d\)) is fulfilled. Then there exist a comparison function \(\psi :[0,\infty )\rightarrow [ 0,\infty )\) and a metric \(\delta \) on \(X\), equivalent with \(d\), such that \(f_{i}:(X,\delta )\rightarrow (X,\delta )\) is a \(\psi \)-contraction for each \(i\in \{1,2,\dots,N\} \) (i.e. \(\delta (f_{i}(x),f_{i}(y))\leq \psi (\delta (x,y))\)) for each \(x,y\in X\)). Moreover, if the metric space \((X,d)\) is complete, then \((X,\delta )\) is complete.''