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Fixed point theory for cyclic \(\varphi\)-contractions. (English) Zbl 1191.54042
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.

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
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