Fixed point theory for cyclic \(\varphi\)-contractions (Q2653940)

So-called cyclic \(\varphi\)-contractions in complete metric spaces, where \(\varphi: \mathbb{R}_+\to\mathbb{R}_+\) is a so-called comparison function, are considered. Some properties of comparison functions are presented. The main result of the paper reads that if \(f: Y\to Y\), where \(Y\) is a complete metric space, is a cyclic \(\varphi\)-contraction where \(\varphi: \mathbb{R}_+\to\mathbb{R}_+\) is a comparison function, then the Picard iteration converges for any starting point from \(Y\). Some further results about the properties of such type of operators are presented in the paper.