Fixed point results for multipoint Kannan-type mappings
Ravindra K. Bisht, Evgeniy Petrov
TL;DR
This work extends fixed-point theory to $n$-point Kannan-type mappings in metric spaces, defining $S(Tx_1,\dots,Tx_n) \le \lambda \sum_{i=1}^n d(x_i, Tx_i)$ with $0 \le \lambda < \frac{n-1}{n}$ and establishing a central fixed-point theorem. It proves that on a complete space with at least $n$ points, such mappings have a fixed point provided there are no periodic points with prime periods in $\{2,\dots,n-1\}$, and at most $n-1$ fixed points in total; a reformulation guarantees the existence of a periodic point with prime period in $\{1,\dots,n-1\}$. The paper also generalizes to $n$-point $G$-Kannan-type mappings (with $G$ in class $\mathcal{G}$) under continuity and asymptotic regularity, yielding fixed points and bounded numbers of fixed points, and shows that weaker continuity notions still preserve results when asymptotic regularity holds. These results broaden multi-point fixed-point theory and link classical Kannan contractions to multi-point and generalized contractive frameworks, with implications for nonlinear analysis and dynamical systems.
Abstract
We introduce and study a new type of mappings in metric spaces termed $n$-point Kannan-type mappings. A fixed-point theorem is proved for these mappings. In general case such mappings are discontinuous in the domain but necessarily continuous at fixed points. Conditions under which usual Kannan mappings and mapping contracting the total pairwise distances between $n$ points are $n$-point Kannan-type mappings are found. It is shown that additional conditions of asymptotic regularity and continuity allow to extend the value of the contraction coefficient in fixed-point theorems for $n$-point Kannan-type mappings.