Advanced Search

W.A. Kirk in 1971 showed that if $T:C\to C$, where $C$ is a closed and convex subset of a Banach space, is $n$-periodic and uniformly $k$-lipschitzian mapping with $k<k_0\left(n\right)$, then $T$ has a fixed point. This result implies estimates of $k_0\left(n\right)$ for natural $n\ge 2$ for the general class of $k$-lipschitzian mappings. In these cases, $k_0\left(n\right)$ are less than or equal to 2. Using very simple method we extend this and later results for a certain subclass of the family of $k$-lipschitzian mappings. In the paper we show that $k_0\left(3\right)>2$ in any Banach space. We also...

Using modified Halpern iterations, by elementary method, we extend and improve results obtained by W.A. Kirk (Proc. Amer. Math. Soc. (1971), 294) and others, which have recently been presented in Chapter 11 of (2001).