General contractions in new type perturbed metric spaces
Bekir Danış
TL;DR
The paper addresses extending fixed point theory to new type perturbed metric spaces by introducing new type perturbed Kannan mappings. It defines the contraction as $ D(Tx,Ty) \le \alpha \bigl[ D(x,Tx) + D(y,Ty) \bigr] $ with $ 0 \le \alpha < \frac{1}{2} $ and uses the exact metric $ d = D/P $ on a complete space. The paper proves existence and uniqueness of a fixed point via a Picard sequence, deriving $ d(x_n,x_{n+1}) $ bounds and showing $d$-convergence to a fixed point. This generalizes Banach-style contractions to a non-continuous, ratio-based metric framework, broadening fixed point applicability in nonlinear analysis.
Abstract
We focus on the new type perturbed metric spaces and introduce a contraction mapping namely new type perturbed Kannan mappings. For these mappings, we show that Banach's fixed point theorem holds. Moreover, this new generalization of Banach's contraction principle does not depend on the continuity of the operator.