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Remarks on fixed points of rotative Lipschitzian mappings. (English) Zbl 1065.47504
Let $$C$$ be a nonempty closed convex subset of a Banach space and $$T\:C\rightarrow C$$ a $$k$$-Lipschitzian rotative mapping (i.e., $$\| Tx-Ty\| \leq k\| x-y\|$$ and $$\| T^n x-x\| \leq a\| x-Tx\|$$ for some real $$a$$, $$k$$ and an integer $$n>a$$). The author extends some results of K. Goebel and M. Koter [Rend. Sem. Mat. Fis. di Milano 51, 145–156 (1981; Zbl 0535.47031)]. He derives very sophisticated estimations of the number $$k$$ depending on $$a\in [0,2)$$ for $$n=2$$ (in the case of a Hilbert space even for $$n=3$$) to prove the existence of a fixed point for the mapping $$T$$. Finally, some applications to Hardy and Sobolev spaces are shown and several open problems are mentioned.
MSC:
 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
Zbl 0535.47031
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