Fixed Point Theorems in n-Point Metric Spaces

Abstract

Classical fixed point theory is based on distances involving two
points. However, numerous modern computational models such as
multi–agent consensus, patch–based image processing, clustering,
and networked systems naturally require distance notions involving
several points simultaneously. In this paper, we introduce and
study n–point metric spaces, a natural generalization of metrics
and $G$–metrics, in which the distance is evaluated on
$n\ge2$ points. Within this framework, we formulate the notion
of an \emph{$n$–point $F$–contraction}, extending Wardowski’s theory
to a genuinely multi–point setting. Our main results establish
existence, uniqueness, and convergence of Picard iterations for
self–mappings defined on $D$–complete $n$–metric spaces.
For $n=2$, the results reduce to Banach and Wardowski’s classical
theorems. Examples, structural insights, and an application to multi–pixel
image denoising demonstrate the relevance of the theory to
nonlinear analysis and computational models.