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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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##### References:
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