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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.

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
Full Text: DOI
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