A new contraction principle on the perimeters of triangles and related results
Tanusri Senapati
TL;DR
The paper introduces a weak contraction principle on triangle perimeters in a metric space, formalized by $d(Tx, Ty) + d(Ty, Tz) + d(Tz, Tx) <= k \, M(d(x', y') + d(y', z') + d(z', x'))$ with $k in (0,1)$, and investigates the associated fixed-point behavior. It proves that, in complete spaces, iterates $x_n = T^n x_0$ converge to a fixed point $x^*$ under $T^2(x) != x$ whenever $Tx != x$, and that $x^*$ is unique if it is not attained by an iterate; it also shows that a weak contraction can have at most two fixed points and is continuous at the limit point. Furthermore, it provides a completeness criterion: the space is complete iff every such weak contraction has a fixed point, offering a novel link between completeness and triangle-perimeter based fixed-point properties. The results extend the landscape beyond classical Banach contractions by focusing on triangle perimeters and perimeter-based dynamics in fixed-point theory.
Abstract
In this article, we introduce a new type of mapping contracting perimeters of triangles in a complete metric space and present related fixed point theorem. We study the metric completeness property of the underlying space in terms of fixed point of our newly introduced mapping. In support of our result, we present several examples.