Remarks on fixed points of rotative Lipschitzian mappings. (English) Zbl 1065.47504

Górnicki, Jarosław

Commentat. Math. Univ. Carol. 40, No. 3, 495-510 (1999).

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.

Reviewer: Slavomír Burýšek (Praha)

MSC:

47H10	Fixed-point theorems
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

rotative mappings; fixed points; Banach spaces

Citations:

Zbl 0535.47031